The Archives are located under Bonus Features in the main menu. Navigate to the Costumes section using and . Costumes are unlocked with Armory Keys which are earned by getting XP and leveling up.
There is one costume for each character for a total of 24 costumes, so it'll require 24 Armory Keys to unlock this achievement. Your 24th card is unlocked at level 49, which means you can have this achievement unlocked by then. See "The Hero We Deserve" for more info on leveling, and see "Hoarder" for more info on the Archives.