To unlock various characters story modes within the game you must first clear at least on other characters Story Mode first.

Unlockable	How to unlock
Alito	Clear Astral's story mode.
Astral	Clear Yuma's story mode.
Dr. Faker	Clear any characters story once.
Konami Kun (The Player Character)	Clear ZEXALs story mode.
Quattro	Clear Shark's story mode.
Tori Medows	Clear any characters story once.
Trey	Clear Yuma's story mode.
Vetrix	Clear any characters story once.
Zexal	Clear Astral's story mode.