To unlock costumes, you can play through either Story Mode or Time Attack. However with Hitomi, Bayman, and Tengu, you have to unlock them with Time Attack. Also, when the costumes say they are unlocked on a certain difficulty; that is the earliest one can unlock them.

Kasumi: C3 - C7 (Easy Mode); C8 - C15 (Normal Mode); C16 - C20 (Hard Mode)
Gen Fu: C3 - C4 (Easy Mode); C5 - C8 (Normal Mode)
Tina: C3 - C7 (Easy Mode); C8 - C14 (Normal Mode); C15 - C16 (Hard Mode)
Zack: C3 - C4 (Easy Mode); C5 - C8 (Normal Mode)
Jann Lee: C3 - C4 (Easy Mode); C5 - C8 (Normal Mode)
Ayane: C3 - C7 (Easy Mode); C8 - C15 (Normal Mode); C16 - C20 (Hard Mode)
Hitomi: C3 - C7 (Easy Mode); C8 - C11 (Normal Mode); C12 - C14 (Hard Mode)
Tengu: C3 (Easy Mode)
Hayabusa: C3 - C4 (Easy Mode); C5 - C9 (Normal Mode)
Helena: C3 - C7 (Easy Mode); C8 - C11 (Normal Mode); C12 - C13 (Hard Mode)
Bass: C3 - C4 (Easy Mode); C5 - C8 (Normal Mode)
Leon: C3 - C4 (Easy Mode); C5 - C7 (Normal Mode)
Lei Fang: C3 - C7 (Easy Mode); C8 - C15 (Normal Mode); C16 - C20 (Hard Mode)
Ein: C3 - C4 (Easy Mode); C5 - C7 (Normal Mode)
Bayman: C3 - C4 (Easy Mode); C5 - C6 (Normal Mode)