Some characters begin the game with some alternate costumes available:

Bishop: 70's costume
Colossus: Xtreme costume 2
Cyclops: Xtreme costume 2
Jean Grey: Dark Phoenix costume
Juggernaut: Original costume
Magneto: Ultimate Non-Helmet costume (starting costume without helmet)
Nightcrawler: Xtreme costume 2
Rogue: Coat costume
Scarlet Witch: 70's costume
Storm: 70's costume
Sunfire: 70's costume
Toad: Brown costume
Wolverine: Ultimate costume

Successfully complete Act 1 to unlock the following costumes:

Colossus: 70's costume
Cyclops: Original costume, 70's costume
Iceman: Snowman costume
Jean Grey: Phoenix costume, 70's costume
Nightcrawler: 70's costume
Storm: 70's costume (another one)
Wolverine: Original costume (Brown and Gold)

Successfully complete Act 2 to unlock the "Age of Apocalypse" costume for all characters that have them.

Successfully complete Act 4 to unlock the following costumes:

Colossus: Xtreme costume, Gladiator costume, Ultimate Non-Steel Skin costume (starting costume with normal skin)
Cyclops: Xtreme costume, Snowfields costume
Iceman: Non-Ice Skin (the same as in X-Men Legends)
Juggernaut: Original Non-Helmet costume
Nightcrawler: Xtreme costume
Storm: Xtreme costume, Snofields costume
Wolverine: Weapon X Experiment costume, Xtreme costume, Snowfields costume

The bonus characters already have the following costumes available:

Deadpool: Unmasked costume
Iron Man: 2099's costume, Age of Apocalypse costume
Professor X: Shadow World costume (the same from X-Men Legends)