He is best known for having created the role of Geoffrey on stage in the Shelagh Delaney play A Taste of Honey, a role which he recreated opposite Rita Tushingham in the 1961 film of the same name. In 1962 he won Best Actor at the Cannes Film Festival for his performance, and was also nominated for the BAFTA "Most Promising Newcomer" award.
Melvin joined Joan Littlewood's Theatre Workshop company at the Theatre Royal Stratford East while still a student. In 1958, he appeared in productions of Brendan Behan's The Hostage. In 1963, he was in the original cast of Oh! What a Lovely War.
More recently, Melvin has returned to the Theatre Royal in Stratford as a trustee and archivist. He also appeared as the sinister Bilis Manger in the Doctor Who spinoff Torchwood.