Martin's father, a singer, and his mother, an actor and extra, set the stage for his early foray into the entertainment industry. By the time he finished school in 1975, Martin had already acted in a few advertisements, but he knew he needed a more stable career. He formed a cabaret act and began performing on a cruise ship at the tender age of 17. Just a year later, in 1976, Martin made his West End theatre debut in Jesus Christ Superstar at the age of 18, having previously trained at the London Academy of Music and Dramatic Art.
Martin's theatre credits continued to grow, with notable roles in The Rocky Horror Show in 1979, where he played the iconic character of Rocky Horror, and Little Shop of Horrors in 1983, in which he brought to life the voice of Audrey II, describing it as a "deep, black, Harlem voice". He also appeared in Time in 1986, 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 addition to his theatre work, Martin has also had a successful career in voice acting. He began providing voices for advertisements and jingles, and in the 1980s, he started working as the voice of Sugar Puffs' mascot, Honey Monster. By the 1990s, Martin had shifted his focus primarily to voice acting, which has led to a wide range of roles in films, television shows, and commercials.
Some of Martin's notable voice acting credits include 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 has also won a Talkie Award for his portrayal of Judge Dredd on BBC Radio 1 and has recorded voiceovers for American and British commercials and film trailers.