***SPOILERS***
This achievement is for beating the final boss, Wheatley.
You start out in a completely black walled room with Wheatley hanging in the middle. A 6-minute timer starts, which is how long you have before you die from the neurotoxins. From playing through the rest of the game, this will seem like an unwinnable situation. Lucky for you, there is a pipe running conversion (white) gel on one of the walls.
Phase 1:When the fight starts, he will start lobbing bombs at you. Hide behind the pipe and one bomb should hit the pipe and break it, sending conversion gel all over the floor and some wall panels. Now you can use portals. He will continue to lob bombs at you, so jump around to get out of their way. Make sure you get him to face so his unprotected back is towards one of the hanging white wall panels. Now, when he throws a few bombs at the same point, drop one portal at that point and one on the wall panel behind him. The bomb will hit him and stun him. At this point, GLaDOS will give you a core. It will be hanging over a catwalk behind Wheatley. Portal up there and grab it. As soon as you do, the catwalk will collapse onto the pipe below it, spilling repulsion (blue) gel around the room. Use that to jump up and stick the core onto the glowing red spot on Wheatley. (it will attach automatically when you are close enough, you don't have to press anything)
Phase 2: His shield will be in a different position this time, and the bombs will launch nonstop. Put a portal directly beneath him. Same as last time, make a bomb land in your other portal, and it will shoot up and stun him again. GLaDOS will hand you a core in the same place as last time, but this time, the catwalk is gone. Portal up to where the catwalk was, then jump down onto the repulsion gel to bounce across to the other unbroken catwalk. Turn around, and grab the swinging core. Same as last time, use the gel on the floor to bounce up and stick it onto the glowing red spot on Wheatley.
Phase 3: This is the same as phase 1. Use a portal on a high wall panel to hit him in the back with a bomb. Now GLaDOS will give you a core on the other side of the room. You will know where by the wall she knocks down. She will be far too high to reach, but luckily, she also broke a pipe, spilling out a path of propulsion (red) gel. The path leads straight to a white wall. Put one portal on that wall, and one directly beneath the core. Then start at the end of the red path, and run all the way down it into the portal. The speed will launch you up high to grab the core. One last time, bounce up and stick it on Wheatley.
Phase 4: After a short scene, the sprinklers will turn on and wash away all the gel except for a small white patch right under Wheatley. You will need to press the stalemate resolution button in the newly revealed room. The only way in is to put one portal on the white patch under Wheatley, and one on the ceiling right above the button. As soon as you jump in though, Wheatley will reveal his last trap, as he blows up the button. Another short scene, and a bit of the ceiling will collapse, showing the night sky, and the moon.
Remember how portals only work on white surfaces?
Good.
Shoot a blue portal at the moon. (thanks to Gen Fire Dragon for testing that it's always the same color)
And brace for the epic ending. (and this achievement)
EDIT: Added video guide. Credit to Silver Scorpian for making and posting it.