Like many minor heroes, Huntress has gone through a few iterations over the years. She began life in What If?-ville as the daughter of Batman and Catwoman (which I guess makes her Batman, Jr's half-sister? Stay gold, Super Sons, stay gold), but eventually made it over into mainline continuity as Helena Bertinelli, mob-daughter-turned-revengeress.
For our Huntress, let's keep the basic origin story: eight year old daughter of a crime boss sees her family murdered by rival criminals, pledges revenge on the whole organization. We'll be keeping her Catholic, but will also be moving her heritage to the cartels of Central America. What can I say? I'm a sucker for Catholic girls. Although, we'll be giving that bared midriff costume thing a miss. I mean, if Batman wears a big symbol on his chest to bad guys shoot at that, rather than his head, a noticeable bare stomach, no matter how good it looks posing, is probably not the smartest tactical move.
Anyways, after Helena Vasquez's family is murdered before her, she is whisked away by loyal family bodyguard Sal. With the bulk of the Vasquez fortune either seized by rivals or locked away in a trust fund that will come to Helena upon turning 21, Sal is forced to do some nasty things to nasty people in order to collect on funds outstanding. Not able to trust anyone with caring for Helena, he's forced to bring her with him as he travels the world collecting on debts. Over time, this becomes his main occupation, one that Helena takes a willing part in.
Helena enjoys tracking down crooks with Sal, that she's doing so at the behest of other crooks doesn't phase her much. Sooner or later they'll piss off the wrong person and will need to be 'dealt with' too. Criminals may be able to avoid the official law, but they can't avoid the law of the jungle. The comeuppance that these people experience at the hands of Sal and Helena are perhaps the only consequences of their criminal lifestyle that they will experience anyways.
This keeps her going for several years, learning the tricks of the trade from Sal (ala The Professional) until Sal himself is betrayed and killed by an employer. Helena, who is now 19, snaps. Her rage is only heightened when it turns out that the hit was put in motion by the same rival crimelord that killed her family. Turns out there is some sort of criminal merger in the works and the new guy in charge is demanding his associates clean up any loose ends. Once she emerges from her initial bloody fugue state and starts to work her way up the criminal ladder, Helena find progress slow going. People just aren't afraid of her, no matter how much she hurts their bottom line.
It's around this time that she hears of Batman (say, .5 SU ago) and his war on crime. Criminals seem really freaked out by the presence of a 'cape' so Helena thinks to mimic the Bat's success on her own. She starts laying plans, assembling connections and resources, in anticipation of her 21st birthday when a large chunk of the remaining Vasquez fortune will become available to her. When she turns 21, she gets herself a few presents - a costume, cutting edge equipment, and revenge on the man who had her family killed.
Fast forward now to Issue #1 of Huntress (#37/52) and Helena Vasquez is continuing her one woman war on crime. For the past several years, she's been focusing on the peripheries of Black Mask's expanding criminal empire. We join her as she comes to Gotham, ready now to strike at the top. This will be easier said than done, of course, as not only will she be facing off against Mask's hired thugs (both human and extra-human), but she'll come into conflict with the Batman himself. He wants Black Mask taken down too, of course, but he also understands that if he does that, the resulting power vacuum would lead to a gangland civil war that would tear Gotham City apart.
At the end of the first arc, Batman will have succeeded in talking some sense into Huntress. "This is my town and you play by my rules. Learn them or leave." Taking that advice to heart, understanding that she knows very little about how Gotham's madhouse of an underworld is set up, Huntress takes on a sidekick of sorts - a young boy named Tommy Monaghan - who has been through some serious crap of his own.