She is a teenager who was murdered by an unknown killer.
Betsy's (unnamed) mother married a man named Craig Wilson. Betsy and her new step-father didn't get along, and they fought constantly. After Betsy was killed while sunbathing, her step-father was the prime suspect in her murder.
Dexter Morgan used Betsy's death to drive a wedge between the police force and Sergeant James Doakes. At the crime scene, Dexter told Doakes that the blood mist on Craig's shirt was from a bludgeoning. Doakes then took Craig in for questioning and accused him of the murder.
Dexter's blood analysis report, which he later placed on Doakes' desk, proved that Craig wasn't the killer and he was exonerated. Doakes realized it was Dexter's plan to make him look incompetent, and it drew his ire.
It is unknown if Betsy's real killer was ever caught.