To be honest, I don’t care much for the Guilty Gear story(and neither should you!), so I’ll just copy/paste some of her background info and add some extra stuff at the end.

Name: ミリア＝レイジ Millia-Rage
Height: 169cm
Bloodtype: B
Weight: 48kg
Birthplace: Russia
Birthday: Orphaned, so unknown
Eye Color: Blue
Likes: Chasing cats
Important Things: Chastity
Dislikes: Zato, fallen-out hair
Voice Actress: Yuko Sumitomo

P.W.A.B. Report #2043: Formerly a hired assassin, the subject is now working with the IPF to dismantle the Guild and capture Zato-One in exchange for her acquittal. The test case for an undisclosed new form of occult art, she uses her hair as a weapon. It is believed her hair falls under the same classification as Gears and Forbidden Beasts. She can be easily controlled if her desire for revenge against Zato is used against her. Although she is not a major threat, her capture and dissection will offer insight into the forces that drive her hair.
Risk Rating: C

Born of Russian descent, Millia lost both of her parents at an early age. She was thereafter adopted into a nearby Assassin Syndicate, where she practiced in the art of Hi-Deigokutsuipou, or the “Six Forbidden Magics”. Using the Sixth Hi-Deigokutsuipou, “Angra”, Millia’s hair gained the abilities she currently possesses. Zato-1 practiced a similar technique, the Fifth Forbidden Magic, “Shokusei Kagejin”, by sacrificing his sense of sight.

Losing both of her parents when she was still a child, Millia was trained by a group of assassins to use her hair as a weapon. However, her conscience would not allow her to murder innocent people, and as a result, she left the organization. Being constantly pursued by the assassins, she decided that it was more efficient to put an end to the organization than to run away from it. She entered the tournament to defeat Zato, the assassins’ leader. Her plan was technically a success and she did gain her freedom, but her feelings-the guilt of causing her former master to be overtaken by a shadow and the fear of having to confront him again-still troubled her. Finally, out of her agony came a conclusion… “I must finish him off, once and for all.”

In the games, her relationship with Zato is a bit vague, as she hates him but seems to pity him at the same time. The GG drama CDs, namely Night of Knives confirm that they used to be lovers when she was in the Assassins Guild. You can read more on their backstory in the Drama CDs section.