You can unlock cotumes by doing various things in the game.

Costume	How to unlock
Armor Spider-man (Black and Greys)	Complete Act 2
Ben Reily	Complete 25 city events
Cosmic Spider-Man	Complete Act 1
Iron Spidey	Collect 150 Spider Tokens
Spider Carnage	Defeat 500 enemies
Spider-Man 2099	Complete 50 city events