My Friendly Neighborhood

This achievement is for beating the final boss of the game, a multi-headed monstrosity composed of several different puppets. Easiest way to do this is using the Conclusion minigun (refer to my achievement guide on how to obtain that). Alternatively, you can stock up on Novelist ammo and Punctuation grenades to kill it. The boss doesn't do anything special besides walk toward you and attempt to damage you just like the normal enemies in the game, but just damaging it enough will stagger it and give you plenty of time to make distance. The achievement will pop once you've triggered the ending cutscene, and to my knowledge you have to do a another playthrough for the other achievement that requires sparing the boss.