The requirement for this achievement is actually a little misleading. You actually need to have 15 Collectables currently on hand, not total. I found this out after buying the Commando Knife for 7 Collectable points and not having the achievement unlock after getting 8 more. Instead the achievement unlocked after I had 15 currently on hand.
In short, make sure not to spend any of your points until you get 15 total first. The collectables are the little floating square items in each level and obviously the ones in Worlds 1 and 2 are the easiest ones to get.