Martin's father was a singer, and his mother was an actor and extra. In 1975, he had acted in a few adverts by the end of school, but needed to get a proper job. He got a cabaret act together and started performing on a cruise ship at 17. In 1976, he made his debut in the West End theatre with Jesus Christ Superstar at 18 after a period of training at the London Academy of Music and Dramatic Art.

He later moved on to other theatre roles in the West End, such as Rocky Horror in The Rocky Horror Show in 1979, the voice of Audrey II in Little Shop of Horrors in 1983, which he described as a "deep, black, Harlem voice"; Time in 1986, Munkustrap in Cats from 1987 to 1990, Miss Saigon in 1989, Joseph and the Amazing Technicolor Dreamcoat in 1990, and Vince Fontaine in Grease in 1993. In 1980, he narrated a trailer for Saturn 3. Martin had been doing jingles and adverts, but started doing voice acting, including providing the voice of Sugar Puffs' mascot Honey Monster from the 1980s onwards. In the 1990s, he shifted to focus more on voice acting.

Martin has provided voices for The Adventures of Mole, The Adventures of Toad, The Neverending Story III, FernGully 2: The Magical Rescue, Corpse Bride, Ultramarines: A Warhammer 40,000 Movie and Minions. He won a Talkie Award for his portrayal of Judge Dredd for BBC Radio 1. He has also recorded voiceovers for American and British commercials and film trailers.