Bat-Girl made her first official appearance in 1961's Batman #139, but the original Bat-Girl is not Barbara Gordon, the person who is most associated with the Batgirl title. The original Bat-Girl was named Bette Kane. Kathy Kane, the original Batwoman, was Bette's aunt.
Over the course of her comics history, Bette has had a couple different looks and identities pre- and Post-Crisis, as well as in Rebirth. Here's a breakdown.
Pre-Crisis: Bat-Girl
Bette Kane appeared as Bat-Girl seven times between her introduction in 1961 and 1964, when she was removed from the rotation in 1964 by Batman editor Julius Schwartz because she was deemed too silly. In those appearances, she worked with her aunt and assisted Batman and Robin. Her original costume was red and green, to coordinate and complement her counterpart, Robin.
Bette and Kathy were originally introduced as potential love interests for Batman and Robin, likely to dispel rumors of the dynamic duo's alleged homosexuality.
Post-Crisis: Flamebird
Mary Elizabeth "Bette" Kane was reintroduced Post-Crisis as Flamebird, a sometimes member of Teen Titans West. This version of Bette Kane was a spoiled Los Angeles debutante with professional-level tennis skills. She decided to be a crime fighter when she witnessed Robin's exploits on the news. She wanted to gain his attention and favor by becoming a hero just like him. That relationship never blossomed in the romantic way she had hoped, which led to her to temporarily quit crime-fighting. During her hiatus, she found that being a superhero gave her greater joy than any tennis championship or perfect grades ever could.
She eventually found her way back to crime-fighting and tried several times to reunite with the Teen Titans West team. While working a mission with Hawk and Dove II, Dove II accurately described Bette as a lonely person desperate for companionship. This aspect of Bette's character helps to explain her often fervent behavior around other heroes and her desire to be a committed member of a crime-fighting team.
Bette went solo for a while after being rejected as an official Teen Titans member. She continued to cross paths with various Titans and other heroes during her solo campaign. For example, she helped Beast Boy clear his name after an impostor framed him for various crimes and during a recruitment party for the ill-fated Titans L.A. team, Flamebird subdued a party-crashing Duela Dent.
Eventually, Bette packed up her life and moved to Gotham City, where she enrolled at Gotham University. There, she crossed her path with her cousin Kate Kane, the current Batwoman. She learned Kate's secret after Kate rescued her from The Cutter, who had kidnapped Bette with intentions of removing her ears in his endeavor to build the "perfect woman" using stolen body parts. At the end of that story, Bette dons her Firebird costume and tells Kate that she desperately wants to team up with her and work together as partners.
DC Rebirth: Hawkfire
In the DC Rebirth timeline, Bette Kane has been embraced by her cousin and the two have become partners, though that relationship is hard-won. Kate tried to stop Bette's crime-fighting career by destroying her Flamebird costume. That did not stop Bette. Bette donned a mask and sweatsuit and continued to hit the streets fighting crime. It was only after seeing Bette's commitment to the cause that Kate eventually relented and trained Bette properly to debut as her partner.
When designing a new costume to replace her destroyed Flamebird one, Bette re-branded herself yet again, this time going by the name Hawkfire. Prior to Batwoman's training, Bette was already an exceptional athlete. Her endurance and strength training from her athletic career lent itself well to crime fighting. Kate elevated Bette's combat skills by training her in various forms of martial arts. Though Bette is quite adept at a variety of martial arts, kick-boxing is her specialty. That skill is especially obvious in her fighting style.