Cosmic Doops are these green ghost things, that look a lot like Slimer from Ghostbusters, IMO. There are regular Doops, but those will not give you this achievement. Cosmic Doops are very big, and quite strong. When they appear, you will see a red marker on your screen, much like you do for a lot of the bosses in the game. It'll have a picture of the Cosmic Doop in it. Follow the arrow to it and kill it.
It also shows up on the minimap, with a blue diamond around it. If you press the right stick in, it superimposes the minimap over your screen and you'll be able to see much farther, and should be able to locate it much easier this way. The fight itself wasn't hard, quite a few people showed up to help and I was playing a ranged character, so I just pewpew'd and dodged the occasional attack until it died.
From what I can tell, it's appearance is entirely random. The one I fought was in the sinister lands in chapter 7, for what it's worth. The Doops and Cosmic Doops drop a lot of stuff, and they often drop legendary marks, so it's worth killing them when you see them.
EDIT : After I left a story mission in chapter 7 there was another one, and there was one in the hydra area in chapter 8. Though with the one in chapter 8 no other players were there, so I ended up having to solo it. Took a decent amount of time and quite a bit of micro but I was able to do it at level 50, so you don't need a group for it.