In which I answer yet another of flatbear’s asks! She’s pretty buried in commission work, and Bobbi is a favorite of mine, so here goes. :)
First, I feel like I should mention that Bobbi is a very problematic character. A lot of her history is defined by her relationship with Clint Barton, and basically every misfortune that a female character in comics can suffer, she’s suffered. She actually started out working with (and dating) Ka-Zar but no one ever really mentions that anymore. She’s also a biologist, which no one really mentions much anymore either.

She joined the Avengers because of her relationship with Hawkeye, which broke down after she got left in the past (!) and raped by Phantom Rider (!!) and then accused of wanting it (!!!) by Hawkeye (!!!!) her husband (!!!!!) who then broke up with her after he found out that she let Phantom Rider fall to his death (!!!!!!) because ~morals (sob). So there’s all that, which most writers tend to gloss over, but which still does get brought up from time to time in a round-about sort of way.
Anyway, she ditched the Avengers for a while and filed for divorce and had a team of her own, but naturally ended up back with the Avengers after she ran into Clint while they were both investigating the (hilarious) Great Lakes Avengers. Eventually, she went back to the West Coast Avengers and got back together with Clint (sigh) with a brief interlude where Ultron used her thought patterns to make himself a creepy robot wife. Anyway, she and Clint decided to scrap the whole divorce thing but then she jumped in front of an energy blast to save his life and died.

Except not really because it was a Skrull and she’d been held captive while a Skrull impersonated her, so when she came back she still divorced Clint. But then they decided to date anyway, except then he started fucking Spider-woman and…
…you see what I mean about her being a problematic character?
In spite of that, though, Bobbi is one of my absolute favorites because she’s tough and sassy and she’s finally gotten to a point in her career as a comic book character where she don’t need no man. She’s also got some sweet super soldier powers (she got shot and they gave her Infinity Formula, short version). I always talk shit about wanting to write a Mockingbird solo book. I just think out of all of Marvel’s characters, she’s the one that most deserves a chance to be her own person.

If you’re interested in reading some good Bobbi stories, I’d hit up Bendis’s run on New Avengers. Start with Volume 10 and just kinda keep going; that will take you through Dark Reign, Seige, and Heroic Age. There’s also Hawkeye & Mockingbird, which I have mixed feelings about, but which is worth a read if you want to know more about the character. She also shows up in Widowmaker, which is great.
Hope that helps you out!