McCoy was born in Dunoon, on the Cowal peninsula, to an Irish mother and English father. He was raised primarily in Dublin. In his youth, he trained for the priesthood, but gave this up and spent time working in the insurance industry. He worked in The Roundhouse box office for a time, where he was discovered by Ken Campbell. McCoy currently resides in his native country of Scotland with his wife and two children.

He came to prominence as a member of the comedy act "The Ken Campbell Roadshow". His best known act was as a stuntman character called "Sylveste McCoy" in a play entitled An Evening with Sylveste McCoy, where his stunts included putting a fork and nails up his nose and stuffing ferrets down his trousers, and setting his head on fire.

As a joke, the programme notes listed Sylveste McCoy as played by "Sylveste McCoy" and, after a reviewer missed the joke and assumed that Sylveste McCoy was a real person, Kent-Smith adopted this as his stage name. He was also reportedly a bodyguard for the Rolling Stones.

Some years later, on discovering that there were thirteen letters in 'Sylveste McCoy' and feeling that this might be unlucky, McCoy added an 'r' to the end of 'Sylveste'. Notable television appearances before he gained the role of the Doctor included roles in Vision On (where he played Pepe, a character who lived in the mirror), an O-Man in Jigsaw and Tiswas. McCoy also portrayed, in one-man shows on the stage, two famous movie comedians: Stan Laurel and Buster Keaton.

He also appeared as Henry "Birdie" Bowers in the 1985 television mini series about Robert Falcon Scott's last Antarctic expedition, The Last Place On Earth.