The idea of getting a barb through hell seems to be a bit of a joke amongst the community. Even after this guide, I still feel like its a joke.
This is a collection of ideas geared towards helping the Barbarian get through nightmare. It is assumed that a strategy for getting through normal is already been executed/known. I have no idea if any of this will be nearly as effective in hell. Good luck to the next poor soul to try this.

Why is the barb so hard?
Barbarian is hard because of how damage is calculated and how it must be dealt. With most classes, you throw a spell out, the spell hits a target, and damage is calculated. With the barb, you are essentially forced to use the mechanic of physically hitting your target. Chance to hit your target is calculated on a attack rating vs defense basis as well as your level compared to mobs. So even though an attack is thrown out, it may not hit and no damage will be calculated.
Barbarian also has virtually no multi-hit ability until level 30. So now we are forced to not only to frequently miss, but to frequently miss only 1 target.
Barb damage is based on the weapon you are using. Loot is quite a bit less reliable as skill points are for other classes.
Barb is forced into melee range to be effective, this often involves getting surrounded and overwhelmed. It also put many mobs into range to attack you, forcing you to take hits.

The build.

Stats
(1a) Strength: “Each point of strength increases your damage by 1% of your weapon's damage.” From what I can tell, this is base damage. This then gets increased by whatever +enhance damage you might have such as mace mastery (not on your weapon, that is for the weapon?) And is also modified by something called “Stat bonus”, based on your weapon (flail) this is str/100. All this then gets modified by skills? Regardless, it seems Strength plays an integral role in damage calculation. Putting 100, 150, maybe even more points here may not be too unreasonable, to counteract the diminishing effects of crushing blow once a mob gets lower on health.
(1b) Dexterity/Attack rating/Level: This stat affects you chance to hit enemies. More dex means more attack rating. Being above or below your opponents level also affects you chance to hit. Your chance to hit seems to formulate out as getting worse as you level up versus mobs of equal level and equal attack rating vs defense. 75% at level 10, about 62% at 30, 58% at level 50, 56% at level 70.
¤Consider a fallen right outside the rogue camp, they are level 36 with 369 armor. Beetles in act 2 start off at level 43 with 887 armor. Venom Lords in the Chaos Sanctuary are level 57 with just under 900 armor. Overleveling and/or getting quite a bit of Dexterity needs to be considered if you want to keep your chance to hit up. You cannot go above 95%.
(1c) Vitality: As you can see, sacrificing large amounts of health my be needed to keep up the pace of killing. Barbs do have high health gain per level to compensate, as well as Battle Orders.

Skills
(2a) Double Swing/Frenzy, seems to be the popular choice for getting a barb anywhere. With fast hits and incredible move speed its easy to see why this might actually be the build of choice. I suggest 8 to 9 points to make double swing cost zero mana, this helps with mana burn mobs. Double swing also hits a lot faster than frenzy, giving more chances at crushing blows. Double swing does not improve base damage very much.
(2b)Frenzy places a buff on you that increases attack speed and more importantly, run speed. This buff can be applied to double swing for insane attack speed. Just tapping a mob to get the frenzy buff and then just running is the premier way for raw movement. Frenzy also has much better damage and attack rating than double swing, but finding enough mana leech (based on damage) to use it exclusively might be an issue.
(2c) Battle Cry. This ability is probably an absolute must at some point. Halving you opponents defense and quartering their damage is too much to ignore. 3 points into the warcry tree is well worth compared to just getting more frenzy or mace mastery. Eventually this is used as a synergy to War Cry, at level 20 monster do 44% less damage.
(2d) Leap/Leap attack: Is one of the best movement abilities in the game! Leaping out of danger and leaping to safety. Leap over gates in the flayer dungeon, leap over the river in act 3 and in act 4, leap over the maggot lair and the arcane sanctuary. Leap attack starts off with a better radius, Leap attack is actually be the skill of choice here when you can get it.
(2e) Mace Mastery: Giving you 5% damage and 8% rating per level as well as some critical strike, this is a slightly better option than most other skills for raising stats. But lacks the utility some skills provide and doesn't provide any synergy's. How to balance how many point to put here is confounding, but at least 1 point for the huge bonus that first point provides.
(2f) Concentrate: With a rhyme shield you no longer can be frozen, making this skill appealing, Toughening you up and being uninterruptable. This skill is the bread and butter of one weapon builds. Keep frenzy on the switch to get some haste, then swing swiftly with concentrate, mebe. 2hander barb?
(2g) Berserk: This skill leaves you vulnerable to taking hits. This also changes you physical damage to magic, not like your expected to do much of that anyway. This skill also increases you attack rating the most of any skill, giving you a better chance to hit. I think you can still crush with this skill...
(2h)Battle Orders/Battle Command/Shout: Command give you +1 to skills which translates into 7 or more points elsewhere depending on what you consider useful in your build. Orders is a nice 40% more health which is useful. Use shout before tough packs and bosses, it helps, a little.

(2i) War Cry: This ability is AMAZING, for the reason that it is a SPELL! Not tethered to a paltry weapon for damage and with 100% accuracy this ability soon makes quick work of the old style of hack and slash combat. War Cry Hits all targets around you, and with the barbarians habit of getting surrounded this means it frequently hits multiple targets, something flailing about can't do. War Cry also stuns monsters around you, making them near useless to even fight you. With these advantages War Cry quickly becomes the dominating spell of choice, keeping up with even the most geared of combat Barbs. The combat tree is changed from a killing style, to a arsenal of niche tools; Concentrate for chunking bosses down with crushing blow, Frenzy for move speed, and leap for getting past the impossible. The damage on this warcry is not as comparable to a javazon or a Sorceress, but it gets by as long as it hits more than one thing. War Crys biggest upset is it's high mana cost of 29 at level 20. Getting the most out of this spell means either finding +mana loot, or sacrificing vitality for energy and then chugging mana potions as this spell drains the barbs tiny mana pool very fast. Also affected by cast speed.

Notable Gear/Crushing blow

(3a) Crushing blow is an insane mechanic that chops down a percentage of a mobs current health. 1/4 health for general mobs, 1/8 for bosses. The more health an enemy has, the more damage a crushing blow will do. As enemies drop in health crushing blow becomes less useful and regular damage needs to take over.
(3b) Flails: Maces/hammers/clubs are used for 50% damage to undead and flails are the fastest of this weapon type, more attacks means more crushing blows. Black Runeword requires some sort of blunt object. Good luck finding a knout.
(3c) Strength: Amn Tir. Upon hitting level 25, the Runeword Strength becomes available. With its hardest component being an Amn rune, it is the most accessible way to get a quick 25% crushing blow. Amn runes can be farmed from nightmare countess, or maybe you can get a lucky drop from normal forge. Consider farming 2 Strengths for 50% crushing blow.
(3d) Blood Gloves. Gather all the rubys you can find, as you will need a perfect one to make a pair of these, save a Nef rune from normal countess runs, and find a jewel. These gloves have 5-10% crushing blow which is useful (lifesteal is nice too). Make them as soon as possible, and continue collecting materials to make more later to roll better/more attributes. At certain item levels (31-50, 51-70) they have an increased chance to roll more modifiers. Note: you might make the gloves and not have the level requirement to use them.
(3e) Black: Thul Io Nef, in 3 socket maces. 40% crushing blow as well as a handful of other useful stats, this weapon is amazing. Finding an Io rune off countess can be tough. It would appear that 2 of these is the dream.
(3d) Myth. Hel Amn Nef in 3 socket armor. Finding a Hel rune off NM countess might be hard, using an Amn rune for this takes away from making a strength, same goes for Nef rune and blood gloves. Its an appealing peace of gear with +2 to skills and some other minor perks.
(3e) Malace: Ith El Eth. Keep this baby around, the 'prevent monster heal' might be useful at some point. Along with the open wounds and -100 armor per hit, might make this good for bosses
(3f) Rhyme: Sheal Eth in a shield. The 'cannot be frozen' attribute is more useful to a barb than any other class. This shield forces you out of a frenzy build. Sheal comes from nightmare countess a or a lucky forge or a fortunate random drop.

¤Mercenary: Your (Act 2) mercenary can be a huge part of your overall damage output. Blessed aim will give you some more chance to hit, something like 5%-10% depending on what you current rating is. Might aura is not recommended, there isnt much damage to might up. Holy freeze slows enemies around you and might make the hoards more manageable. I cant think of anyway to make the other Mercs useful.
¤If you can find a 5 socket partisan (ethereal would be super) consider gathering runes for an 'Honer' (Amn El Ith Tir Sol). This runeword is about a reliable as you might get for consistent damage. Otherwise, find another Amn rune and get him a Strength as well. After these, good luck finding the Husoldal Evo, or some other random unique polearm/spear...
¤As for armor for this guy, there isn't much. 'Smoke' would be cool if you find a Lum rune. If you get crazy lucky, an Um rune can make a 'Duress', a Mal rune can make a 'Prudence'. There are numerous unique chests your merc could use. You might want to keep most of these options for yourself over your stealth, and again, good luck finding them. But even with these armors, The Merc is still pretty squishy and easily beaten down by aggressive mobs.
¤Hats are in even shorter supply, a Vampire Gaze would be the dream, but you might have to settle with whatever you find, or some socketed helm with Ral or Ort runes for resists.
¤Overall the Merc is super useful, but hard to gear out. He is hard to keep alive and requires frequent running away to do so. He also starts becoming expensive in the later stages. He also tapers off hard, doing less damage and getting killed faster. Mercs do less damage against bosses and get wrecked by them too.
Other mercs: The rogue (A1) doesn't do much damage, doesn't buff you and is constantly running away to get space from targets, also exceeding squishy. She casts Inner sight to reduce targets defense, but that's about it. The Iron Wolf (A3) merc I don't think does that much damage, he is affected by faster cast rate (stealth) and could help freeze targets, but he also has a tendency to run away and get space. Barb merc (A5), this guy might do some damage, but he's going to steal your weapons to do it.

The Route

¤Assuming you farmed in some way to get to 25, going straight to nightmare after that might not be the thing to do. Consider farming in Normal longer to get more levels, find more gear. Farming Mephisto is an idea, but in no way reliable. Cows? Canyon of the magi?
¤Farming the nightmare Countess will likely be a must with all the runewords stated above. This could be completed in just a few painful runs, it could take hours of failed attempts.
¤Whether this is a 'kill as you go', or a 'kill now move later' run is up to you. The experience gained from NM Countess runs is also something to consider and might be all the grind you need, proceeding to leaping over everything in the trails ahead. Killing stuff in further acts gets hard and slow, mostly due to your level falling off and getting hit penalties from it, in addition monsters just get stronger.

Other Ideas, Final thoughts

(4a) Life steal/Life Tap: What little life steal you might find is halved in nightmare, some mobs and most bosses have additional leech resistance. Life tap provides an additional 50% life steal against mobs cursed with it. Consider finding a life tap wand at some point, possibly for ancients, or bosses (Baal) in general.
(4b)Open wounds: This is a 'bleed' that makes your opponent take damage over 8 seconds depending on what level you are. At level 30 the bleed will do 340 damage, at 50 it will do around 800. I don't know if this damage can be reduced in any way. In addition bleeds stack and are individually tracked. This can be found on your malice, as well as select pieces of gear if you find them. Damage is also halved on bosses apparently. While wounds are active, monster will not passively heal (prevent monster heal)
(4c) Ignore Target Defense: This mod is exceptionally rare. It drops the mobs armor rating to 0, but your level is still rolled into the equation, useful, but not as much as it may seem.
(4d) Prevent Monster Heal: On your Malice, remember to tap bosses with this, and it will probably save you some grief.
(4e) Amplify damage: Amping what little physical damage you do may come in handy and might be overrated at the same time. Found on wands sold from vendors.
(4f) 'Honer' Runeword for yourself. I actually got one of these babies in a knout for my personal attempt, I don't feel like it did much over a weapon with crushing blow.
(4g) Whirlwind: As much as this skill can hit multiple targets, it makes your already poor damage worse for most of its existence. I cant vouch for this skill.
(4h) Grim ward: just because monsters are running around doesn't mean they're dead. I cant see this being great. But it might help you pass tough packs.
(4i) Other Cool Loot: Hawkmail, Rattlecage, Duriel's Shell, Goblin Toe, Gore Riders, Crescent Moon Runeword (Um), Sureshrill Frost, Fleshrender, Baezil's Vortex, Arreats Face, maybe a few more...

I think I'm out of ideas on how to help out the joke that is the Barbarian class. As much as you might think all this executed together might make the run easier, it really doesn't. Stuff is always hard to hit, damage is always low, and getting swamped happens often.

Sources: Myself and the years I've been playing this game. The community and all the ideas we bounce around. Numerous websites like Arreat Summit and various 'wikis' for hard numbers. And a Barbarian of mine that that is currently stuck at the Chaos Sanctuary.

Interesting thoughts. I wish I had time to theorycraft on barb and actually run it.

I may add a few thoughts:

1) Barbs can't actually skip a bunch of acts like other classes. Mobs HP will get higher and higher, and your chance to hit/survivability will get lower. So, you need to improve your gear as you advance, act by act. Items with Fast Ht Recovery, high defense (yeah, defense matters for barbs, improved with Shout/Iron Skin) and some resistances are very important. Note, consider this with the offensive items you quoted.

2) The leveling/farming strategy is quite different of other classes when going solo after Normal:
- 21/22 (ending Baal Normal with current speedrun strategy), farm Baal Normal until 30 or so. Reason: for a Barb, you'll be underleveled and undergeared for Nightmare. Baal minions provide great XP and Baal himself provides good gear. Low tier sets can provide a huge boost (Sigons, Angelic)
- 30ish, runes farm. Here we basically look for the runes to get the really useful low runewords. Amn(11)/Sol(12)/Shael(13) can drop on NM Countess and Normal Cow Level.
- 34ish to 40ish: run through NM. As skipping acts is not a reliable strategy, killing elites along the way is the best way to level until 40. Killing Mephisto a few times can provide some good gear. If you can't kill elites (or even trash mobs), then you need better gear. Some more Normal Baal.
- 40ish to 55ish: Baal NM. Again, this is the best place to get XP and gear. NM Mephisto can be done a few times too, for items.
- 55ish to the end: Run Hell. At this point, reliable upgrades are really a long shot. NM Baal can provide enough survivability (sinergizing with your Warcries) to face enemies, but you will still rely on crushing blow/unique items to deal damage. If you have enough damage, good, if not, NM Baal is the only option to farm. Just like NM, if you can't kill elites, then you are undergeared. Using some time to farm is better than run through the Sanctuary without killing anyone.

3) A few considerations:
- Stealth isn't very good after normal. Without resistances, you are an walking target. Yellow (rare) armors with resistances/FHR/high defense are more useful than Stealth.
- Charms help a lot. Any Cow Level is great to farm resistance/life charms.

Update
Blood gloves may be overrated, after rolling almost 50 of them, best I have found is 30 to a resist and the ever so rare 20% attack speed. But still better than nothing in most cases, Sanders Taboo?

If you hit chance is not up to snuff, and killing anything seems impossible, leveling up to remedy this helps, but is slow, up to you how long you want to spend doing this.
20something to 30 - somewhere in normal, i liked mephy, because loot. 30 to 40 NM countess runs as stated before. 40 to 50, NM Andy runs, loot here too. 50 to 60 could be NM mephy runs, but you might be able to clear all the way to Baal, but it'll be tough.
@AlanVault: 40-55 as a raw level grind does not seem doable in the worldstone keeps, as mobs at the end of act 5 are all level 60+.

Sapphire Crafted Items, known as 'hit power' have the attribute Frost nova when struck. This is useful to say the least to a barb who is being surrounded. Notably the amulet (thul) and the ring (Amn) are easy to equip. They might roll garbage, but better than what you might have.

Edits
Added (#s) at the beginning of sections for easier readability. This guide looked so nice in word, speedruns formatting is limited.
Removed some clutter.
added 'item' in the blood gloves section, not to be confused with character level.