The Batwoman is actually Kate Kane, a socialite whose dismissal from the Armed Services has not slowed her personal war.
Kate Kane was born the daughter of a famous and influential colonel in the US military, a member of military intelligence. She grew up happily, playing with her twin sister in their fairly luxurious family home--her father was one of "those" Kanes though he was not in line for the full weight of the family fortune.
When Kate was still quite young, she, her sister, and her mother were kidnapped by terrorists seeking to ransom Kane children and strike at the US military in one blow. A rescue was staged, and Kate's sister and mother were killed. Kate suppressed the memory for many years. Her father remarried, to a wealthy socialite who wanted his family's connections. Kate and her stepmother are not friends, though they do not fight.
When she tried to serve her country to fight against those who had hurt her, she was ejected for being gay. After a period of dissolute aimlessness, she found a new identity in taking the fight into her own hands, on the streets, as the Batwoman. She didn't ask permission for the cowl, but she hasn't faced any objections yet.
The past refuses to stay buried, however, and shortly after Batwoman emerged onto the streets of Gotham, she faced the challenge of 'Alice', a killer who seemed to know her as she knew herself--because she was herself. Almost. Kate Kane's twin sister somhow survived the terrorist attack, unknown to Kate, and possibly unknown to her father. Batwoman battled with Alice in a feud that ended with Alice falling from an airplane, seemingly again to her death.
Kate blames her father for this, and has ceased speaking to him.
Kate Kane is a conflicted, angry, driven young woman. She lost two of the most precious people in the world to her at a young age, but she was able to channel her rage at this injustice into a desire to serve righteous order. She wanted to be a marine, like her father. This was denied to her due to the military's policy of Don't Ask, Don't Tell, which was a powerful blow to her sense of self. She was too bound by her oath to lie in order to preserve her possible military career, but she still felt the draw to defend, to set things right that had been made wrong.
This drive for justice is powerful, and Kate has encountered several ways in which the laws may fail even the lawful. Bad things happen to good people, and often through indifference--this is something that she must change, that she can change, herself, personally. Kate's desire for control and direction pushed her finally into vigilantism and the carefully orchestrated violence of the hero in the shadows. She took her name from the Bat, as there was a -man, so could there be a -woman.
All of this purpose from the shadows has made it difficult for her to stay connected to Kate Kane as a person, and a long string of broken relationships and poorly-mended friendships testifies to that. She tends to react to failure or frustration by taking her anger out on others, and this has lead to her current state of estrangement from her father. The loss of her sister and mother is an aching wound that will not heal without family's help, but she pushes Jake Kane away, blaming him for keeping her sister's survival a secret.