Neon White

Just like the similar achievement for beating Clocktower before Green reaches the top, this achievement requires you to beat The Third Temple before the final phase. To do this, you have to damage him at every opportunity in-between damage phases (where you stun him and deal a bunch of damage). The most important, and also most difficult time to do this is when you're using Dominion to grapple between crystals. Fire a rocket at Green right before your grapple fully pulls you to the crystal, and if you time your rockets just right, they'll land right as you touch the crystal and break his shield for a brief second. You can very easily get damage in at the start of the level with Fireball and Stomp as well, following the same principle as explained above. If you do it right, you can even kill him before the second-to-last damage phase.

Just keep trying and you'll get it eventually. The hardest part is timing the rockets, but practice makes perfect.