Garbage In, Garbage Out

You need to roll over or incinerate every single garbage before boss makes new ones. You may need additional healers to complete this achievement since burning trash gives a nasty stacking debuff for each burning trash.

You have to clear ALL of the trash EVERY time he throws out trash! It must all be gone or on fire before he throws out more.

Clear it however you can with fire and trash balls. The buff he gets when all trash is cleared is called "Impressed" and looks like a yellow happy mask.
Track the achievement so you can see it on your quest bar area. It will turn from red to white if you've cleared all the trash needed for that round; he will also get the buff.
So EVERY time trash comes out, you need to clear all of it, turn the text white, and give him the buff all over again before the next round of trash comes out.
He has to have the buff when he dies.
If you miss one round, you will have to start over! (At least, that's what happened to us.)

If it looks like you got all the trash, but he doesn't get the buff, it's probably best to start over.

Good luck!

We were having a lot of trouble killing him in time since he does the garbage spawn faster and faster after 50%. This may be obvious to you, but it took us a few pulls to realize we should lust after he hits 50% and lands.