Moore was born in Brixton, London, the son of Mary Elisabeth (née Bruce-Anderson) and solicitor Stanley Moore. He is known for his distinctive speaking voice in a wide range of roles. He has been married four times and is the father of former EastEnders and Family Affairs actress Robyn Moore.
His first radio appearance was probably the Peter Tinniswood series Hardluck Hall in 1964. His most famous radio role was as Marvin the Paranoid Android in the original The Hitchhiker's Guide to the Galaxy series in 1978, a role he has reprised several times. Other notable performances include George Smith in Paradise Lost in Cyberspace, Professor Calculus in The Adventures of Tintin and Harry in Underworld.