Gen Fu
You must complete Eliot's Story Mode.

Leon
You must complete Zack's Story Mode.

Ein
You must complete Hitomi's Story Mode.

Helena
You must complete the Story Modes of all initially available characters.

SPARTAN-458
You must complete Helena's Story Mode.
Tengu
You must complete the Time Attach Modes of all the other 21 characters.

Biolab Core
Completing Story Mode once will unlock the Biolab core Stage.

Nassau
The Nassau Station stage will become available when the SPARTAN-458 character is unlocked.

Tatami
Complete the story mode for all 17 characters. Unlock all hidden costumes for all 21 costumes. Unlock the system voice for all 22 characters. Note this stage is already available in Sparring mode -- this unlocks the stage for other modes.

Movie Theatre Mode
Beat story mode once.

System Voices
Complete a character's Exercise session in Sparring Mode. You can then go choose the voice heard during matches in the Settings menu.

Alpha-152's System Voice
Unlock the System voices for all 21 characters (Ein doesn't have one).