Skirton was born in Bath. He started his career at Bath City. He made 144 league appearances for the club, scoring 44 goals. He joined Arsenal in 1959, playing his first game for the side the next year. He was the club's leading goalscorer in the First Division during the 1961–62 season, with 19 goals. He was nicknamed "The Highbury Express" during his time at Arsenal. After 154 total games for the Gunners and 54 goals, he joined Blackpool in 1966. He later played for Bristol City, Torquay United, Durban City and Weymouth.