He trained at the RADA, and made his acting debut in Enid Bagnold"s National Velvet in 1946. Initially a stage manager and understudy at the Whitehall Theatre for the first two years of Reluctant Heroes, the first Whitehall farce, he subsequently spent a few years in weekly representative before returning to Brian Rix"s company with his first play. Dry Rot (1954), which is about dishonest bookmakers, had a four year run with 1,475 performances.
Ray Cooney joined the cast in 1956 and first met the author at this time.
Chapman followed Dry Rot with Simple Spymen (1958), which was staged 1,404 times over a three year run. Before the production of Simple Spymen closed, Chapman and Cooney had begun their collaboration.
Together they wrote Not Now, Darling (1967, which Chapman adapted for the film version), Move Over Mrs Markham (1968), My Giddy Aunt (1968) and There Goes the Bride (1973). Meanwhile, he also wrote extensively for television including episodes of the sitcoms Hugh and I (1962-1965) and Happy Ever After (1974-1977), both of which were British Broadcasting Corporation vehicles for Terry Scott.
Fresh Fields (1984-1986), for Thames Television, featured Anton Rodgers and Julia McKenzie in the leads.
Chapman was married to actress Betty Impey, from Whitehall and they had four children, Mark, Adam, Justin, and Guy (who died when he was young). Chapman died in Périgueux, France on 3rd September, 2001, aged 74.