This is very simple. As you advance in level you will see 'Archive Armory Key's unlocked (not every level, but fairly regularly). These are not to be confused with 'Archive Access Card's.
When you get an Armory key, you can go into the Archives (located in the Bonus menu), scroll over to costumes, and use this key to unlock one of the 24 costumes available (1 per character) in the chart. Once you unlock them all, the achievement is yours!
You will get the 24th key (and thus be able to complete the achievement) by level 51/52.