In the script of Star Trek, Winona Kirk was referred to as a twenty-five year-old as of 2233, suggesting she was born in 2208. 
Star Trek writers Alex Kurtzman and Roberto Orci revealed Winona, at least as they thought of her, was a Starfleet officer, hence her presence aboard the Kelvin. 
She may have a brother, since James T. Kirk said he was staying at his uncle's farm in Star Trek Generations. It was never made clear which of his parents had a sibling, though, or if he was just using the term for a long-time family friend. In Star Trek, an uncle Frank was cut from the script and reworked into Winona's new husband heard in the film.
According to Time for Yesterday, she died in a fire on the family farm in Riverside, Iowa on June 2, 2283. She was cremated and her ashes were placed in a crypt in a chapel at her hometown of Riverside.