When the indicated requirement is met the corresponding costume will become unlocked.

Adam West:
PlayStation exclusive pre-order bonus; available to ALL Season Pass owners in Fall 2015.

Anime Batman:
Sign into the game with a WBPlay account.

Batman 1st Appearance:
Amazon pre-order bonus; or included in Serious Edition.

Batsuit v.7.43:
Available by default.

Batsuit v.8.03:
Progress through the storyline.

Flashpoint:
Included in Premium Edition or ALL Season Pass.

New 52:
Free DLC costume.

The Dark Knight Returns:
Included in Gotham's Future Skin Pack on Steam.