Keys are found in every main level of the game, and have to be acquired in order to use the exit door and progress in the game. Simply collecting a key is enough to count towards this achievement, regardless of whether you finish the level or not.*
You can keep track of this in the pause menu, under stats. It's the number next to the icon of the key. (Who would have thought).
*From the comments: "Its not true (maybe it was patched) that just picking up the key counts, you need to finish the level. If you quit after picking up the key the stat resets."
"Counter said 116 keys, no achievement. Exited to home, quit, restarted, achievement popped straight away"
Thanks guys :-)