The current Huntress is the third DC character to have gone under that name. There was a character called the Huntress in the forties (first appearance in Sensation Comics #68 1947) but I'm not aware of her ever having had anything to do with Batman.

The Huntress' involvement with the Bat Family starts in the seventies. The Helena Wayne version of the Huntress first appeared in All Star Comics #69 (1978) a comic which starred the Justice Society, the classic Golden Age team. By this time DC had decided that all of their Golden Age characters lived on Earth 2 with their modern counterparts on Earth 1. This meant that the Golden Age versions of the heroes would be older than the modern versions; hence the Huntress was the daughter of Bruce Wayne and Selina Kyle.

The Batman Family comic took place in regular continuity so when DC decided to tell Huntress stories there they moved her to Earth 1. The Huntress continued to appear in backup stories for several years.

When Crisis on Infinite Earths happened Helena Wayne disappeared along with Earth 2. The powers that be obviously decided that the character was popular enough to not only be brought back but to be given her own series.

"It's not strong people who need costumes...it's the weak ones. Weak people have to hide behind spooky capes and masks...make themselves more fearsome...so no one will see them for the harmless creatures they are." The Huntress, The Huntress #2

Helena Bertinelli was the daughter of a Mafia family and the sole survivor of a attack during a family celebration. Fearing for her life she fled with her bodyguard who trained her to protect herself. The ongoing Huntress series tried to distance her from the tag of being a female Batman by focussing on realistic crime such as the Mafia, gangs, etc. The series is unusual in that issues 1 - 6 were published under the Comics Code while all the rest were labelled For Mature Readers.

Helena Bertinelli: "Throughout the whole experience...I kept hoping that Batman would accept me. I guess it's foolish, but I kept thinking that maybe he'd grow to like me, see me as a kindred spirit. I half expected him to say "Why I'd be proud to have you as my daughter..." or something equally cornball. Maybe because my own family was such a failure, I guess a part of me wanted him to be in my family."The Huntress #19

The Huntress was also a member of the Justice League for a time during and just after her own series was cancelled at #19. Since then she has gradually moved into the Bat titles, which meant that she had also moved to Gotham, and somewhere along the line she became a schoolteacher.. Batman never approved of her violent methods trying to discourage her from operating in his town. Other members of the family don't hold to the same negative opinions as the bat though; Robin has always seemed to get along well with her and Nightwing got very close to her (see the Nightwing/Huntress mini-series for the full details).

Batman proposed the Huntress for membership of the JLA the for which was given was that it would show her how to be a better i.e. less ruthless hero, although it was probably done because Batman would be occupied with No Man's Land and the editors still wanted the JLA to have some kind of Bat interest.

The Huntress played an unexpected part in No Man's Land. Huntress starred in the Fear of Faith storyline, but when a new Batgirl appeared few people would have guessed that it was actually The Huntress in a different costume. No Man's Land #0, which was published near the end of 1999, showed that as soon as Batman saw this new Batgirl he immediately recognised her as The Huntress from the way in which she moved. Unbeknownst to the new Batgirl Batman set her a series of tests: protecting part of his territory from Black Mask and his gang; not following Batman; and following his orders.

Batman: "Keep the tower safe."
Batgirl: "I could use a little help."
Batman: "I'm needed elsewhere. You're on this alone. Protect the tower. And try not to be seen."
Batgirl: "I can do one or the other, not both."
Batman: "Then you're not worthy of the mantle of course, you could prove me wrong--." Batman #565

The Huntress as Batgirl managed to protect the cops and the building and that they were supposed to be protecting but not without being seen. By Barbara Gordon.

Batman: "Don't ever follow me again, understand? You do what I say when I say it. Otherwise you're out." Detective Comics #732

Huntress may not have been accepted as the new Batgirl but she showed just what she was made of at the end of NML in Endgame. Joker attacked camp of the rogue cop Pettit and his men tricking Pettit into killing his own men by dressing them up like the Joker, leaving only the Huntress to protect the babies from the Joker and his goons. Batman and Nightwing arrived just in time to get her to Leslie Thompkins hospital.

Huntress: "Happy now...?"
Batman: "Rest. Good work, Huntress."
Nightwing: "Congratulations. That's his highest praise. Honest." Detective Comics #741, No Man's Land: Endgame part 3

Despite all her good work Batman still kicked her out of the JLA at the end of World War III because she may have killed Prometheus.

Huntress clearly still wants to be accepted by Batman but at least she has the respect of both Robin and Nightwing. When Robin says that he knows who Nightwing will be kissing when the bells ring in the New Year, his and the reader's expectation is that he means Babs but he actually goes to The Huntress.

Greg Rucka is currently rewriting The Huntress' origin in Batman/Huntress: Cry for Blood and it remains to be seen how this will affect her but I wouldn't expect her to be accepted into the family anytime soon. Not being in the family means that The Huntress does not know the real identities of any of the Bat Family.