Young Morrissey grew up in Knotty Ash near Liverpool and started acting at the Everyman Youth Theatre. He had his first television role at age 19 in the television series One Summer in 1983.
He became well known for his supporting film roles in The Other Boleyn Girl and The Reaping.
He got married to his longtime girlfriend Esther Freud in 2006; the couple had three children named Albie, Anna and Gene. His grandfather-in-law was the renowned psychologist Sigmund Freud and his father-in-law was the visual artist Lucian Freud.
He acted alongside Emily Watson in the 1998 biographical drama Hilary and Jackie.