You need very quick hits on the first 3 phases to make the 4th possible, as the first balloon will block your jump if you're too slow. The third balloon prevents an even faster 4th hit by being right on the edge of the platform when you'd ideally reach it, so this seems to be the limit of this fight. You have to delay your jump between platforms 2 and 3 so you'll grab the edge of platform 3 after the balloon passes, but before the next laser.

The skulls in phase 4 aren't affected by the global timer, so you can do the same thing every time. When the second "chain" of skulls begins (inside the last large gap before the gaps shrink) I do a medium-delayed double jump followed by a quick double jump to land in the smaller gaps and get onto the first balloon platform. This took a while to get used to, since you can't see the gaps until right before you land in them. You have to know they're coming and stagger your double jumps in the same way.

I'm not sure if grabbing the second key earlier in phase 5 could speed it up any. It'd be a very small time save if so.