Neon White

PSA: It’s come to my attention a couple days after writing this that earning this achievement is more or less entirely dependent on how many pistol shots you can get in on Green during the intermission periods at the start of the fight, i.e. before the first stagger window available. You’ll need around 10 to get an early kill. It’s possible you could also cheese this by playing one of Mikey’s Rushes and simply shooting Green with two rockets at the start of the fight. I’m keeping the guide largely as is despite all of that, but just keep it in mind while going for this.
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A FYI before I begin with the rundown, this achievement is a bit of a pita. It’s not especially hard if you know what you’re doing, but it’s still fairly easy to fail attempts with Green only having a smidge of health left. Might be worth muting the volume so you don’t have to constantly hear “Time’s ticking, White!” while doing attempts; it gets a bit grating after a while.
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You’ll want to start the fight walking towards the inital string of cards (two pistols and one ammo) while looking up at the sky for Green to appear. Get ~3-5 pistol shots in before he puts up his shield and moves to the left (make sure you count the number of ones you saw on Green and keep it in your mind). Move to the right instead and use jump + a charge of Elevate to get to the right platform, then turn back around to shoot the crystals spawned to the left of where you started. This will give you a good window to get in another 2 or so shots. The next crystal sequence plays out as normal, followed by trying to get as many shots in as you can while he moves to the arena-esque part of the level; just be wary of your pistol ammo here so you’re not forced to restart (if your one count is lower than 10 at this point, you should restart). The first proper crystal ‘phase’ plays out completely as normal. Only tips to keep in mind would be to not move to Green’s right: for whatever reason he doesn’t take damage when you are there at this part of the fight. When you get your next opportunity to damage Green, shoot the machine gun for 2-3 seconds then shoot two grenades at him. At the final valid damage phase for this achievement, leave the crystal to Green’s upper-left for last. When you’re ready to shoot it, lob a grenade at Green, then immediately shoot the crystal. If you’ve done all of this properly you should be able to just fire away from there and that’ll finish him off. The achievement will pop while Green falls to the ground for you to do the final blow.

Video demonstration by xl MrProdigy lx:

This is during your first encounter with Green in level 4-3

The trick for this is to shoot him whenever his shield is down not just when he is staggered. As soon as the level starts be sure to grab the gun and shoot him before he spawns the first crystals for some easy damage. After that make sure you shoot him whenever you destroy a set of crystals and he starts to move to the next location.