Not much is known about Abigail's life prior to the events of the game. Its known that she was the assistant of TV producer Frank Mathers, and that he abandoned his current wife and 5 children to be with her.
Events of GTA V
Abigail can be found at the Sonar Collections Dock. To activate her mission the player must purchase the docks for $250,000.
When Michael first met her, Abigail talks about the recent death of her husband, that drowned while he was testing a submarine. Michael mentions rumors that someone tampered with the airlock, which Abigail quickly belies, saying that was never proved. She then asks Michael to find all the pieces of the submarine, so she can prove that Frank is dead and collect his life insurance, and also be able to "sleep much easier".
When Michael comes back to her with all the Submarine Parts, she's surprised to find a piece of evidence that suggests sabotage, she would claim that the police are in on his death, and "intends to take her findings to the FIB." She rewards Michael with a mere $10 and a rare signed autograph from her husband, much to Michael's frustration. The player then has the option of killing her or just letting her go. As you pursue her, she'll still claim that she didn't kill her husband. But if the player cautiously follows her to the city, without ramming her, thus triggering a chase, her destination won't be the FIB, but rather the Airport, where she's more likely to skip town. At this point she'll get out on foot, and scream for help, claiming that you're harassing her, thus, triggering a 3-star wanted level police chase on you.