The optimal way of making record in zombie challenge mode is:

1. Wait untill Jeb spawns, then survive untill he runs out of ammo.
2. Use shotguns what are spawning on level. Do not shoot Jeb, or else he will respawn with full ammo; and he is too tough to kill before he opens fire. Instead shoot zombies.
3. Carry 1 shotgun on back and another in hand, grab new shotguns as they spawn (the new one starts respawning only as the current shotgun on floor is taken), and throw old, but still loaded shotgun to ground so you can take it later.
4. Shoot zombies before they can reach you (but to so in relatively close range, so you can one or two shot them). Throw empty shotguns and melee weapons if you want. Make sure you have ammo. Use the backup shotgun if current one is empty. Take new guns ASAP. I managed to make more then 773 kills that way.

Sorry for the fact what i don't know how to place the proof picture there. Or make the run videos.