Originally from the Halliwell area of Bolton, Lancashire, England he attended Sharples School from 1980-85. After completing his schooling his mother made him attend a play and this sparked his interest in acting. He is a fan of the Smiths and long time supporter of Bolton Wanderers. He is married to Lesley Sharp. They have two sons and reside in London.
Gleaves has been in many television shows since 1992. he was Rick Powell in the television drama series Playing the Field. He had the role of Ray Fairburn in Conviction. Conviction was filmed at Salford Lads' Club which delighted Gleaves because of the building's connection to the Smiths. In The Chase he was the father Tom Bedford. In 2009 he said the idea for Survivors, in which he played the scientist Whitaker, was "great". He played DS Andy Roper in the first two series of Scott & Bailey. Of this role he said " I’ve done quite a few cop shows and as much as I enjoy testosterone, shouting at villains and all that, I really fell in love with this. As soon as I read it I saw something new and original. That was one of the reasons why I wanted to be involved because it was so different and very believable."
His first theatre part was as an extra in Don Carlos at the Royal Exchange Theatre in Manchester. He did several plays there including the lead in Macbeth. When interviewed about this play he said he never had the superstition that surrounded it, "...all that spitting and spinning round — it all seems like quite hard work".