She studied at the Manchester Polytechnic School of Theatre and her first television appearance was a small part in an episode of A Touch of Frost. She went on to star in the acclaimed drama series Band of Gold before landing a role in the film Girls Night alongside Julie Walters and Brenda Blethyn.
In 1999 she appeared in the Victoria Wood penned sitcom Dinnerladies and had a small part in the Johnny Depp/Cate Blanchett movie The Man Who Cried. It was after filming that role in 2000 that she was cast as Eileen Grimshaw in the top rated ITV1 soap opera Coronation Street. Cleaver won praise for her portrayal of Eileen, the mother of a son - Todd - who struggles with his sexuality and announces he is gay. The character of Todd has since left the show but Eileen remains one of the more popular characters among fans.
In February 2008, Cleaver was interviewed by George Stroumboulopoulos on CBC's The Hour while the show was touring the UK, and visited the set of Coronation Street. In a segment during the show, Sue recounted a night from her past where she, partly due to intoxication, ended up spending an undisclosed amount of counterfeit money at a bar one night, and proudly proclaimed she has yet to be arrested for it.
She has type 2 diabetes.