Makepeace was born in Montreal, Quebec, the son of Doreen and Harry Makepeace. He began his acting career in a 1974 Canadian television special, The Ottawa Valley. His next role was in a 1979 Canadian comedy, Meatballs, in which he starred opposite Bill Murray, playing one of the teenage attendees of a summer camp. The film grossed more than $43 million at the box office, and Makepeace was then cast in the lead role in the Fox film, My Bodyguard, which was released in July 1980 to positive reviews, grossing $22.5 million domestically. In his review of the film, critic Roger Ebert said that Makepeace's performance resulted in "one of the most engaging teenage characters I've seen in the movies in a long time." Makepeace also starred opposite Lee Majors and Burgess Meredith in the 1981 film The Last Chase. Makepeace's subsequent roles included appearances in made-for-television films and a few more feature films, including The Falcon and the Snowman in 1985 and 1986's Vamp. He has not appeared in any films released since 2001, moving instead behind the camera as an assistant Director.

In 1981, Makepeace recorded spoken dialogue for the KISS album Music from "The Elder" with Producer Bob Ezrin, but it was not used in the final mix; plans to turn the album into a feature film never materialized.