Beating a character's story mode would unlock their deck recipes and deck edit option in the free duel mode.

Deck	How to unlock
Alito's deck recipe	Beat Alito's story.
Astral's deck recipe	Beat Astral's story.
Deck recipe and deck edit for other characters	Beat Konami-kun's story.
Dr. Faker's deck recipe	Beat Dr. Faker's story.
Kite's deck recipe	Beat Kite's story.
Quattro's deck recipe	Beat Quattro's story.
Shark's deck recipe	Beat Shark's story.
Tori's deck recipe	Beat Tori's story.
Trey's deck recipe	Beat Trey's story.
Vetrix's deck recipe	Beat Vetrix's story.
Yuma's deck recipe	Beat Yuma's story.
Zexal's deck recipe	Beat Zexal's story.