Burnley-born Cheesebrough started his career with his hometown club before City snapped-up his services in 1959, making his debut in August of that year at West Ham United.

Cheesebrough could operate in attack, just off the attack, or on the wing and often moved throughout these positions both at City and at his time at Burnley. His most impressive games, however, came out wide for City during their run to the FA Cup final in 1961. He'd won an England U23 cap and impressed the Leicester scouts in 1957 when he netted a hat-trick against them.

His time at Filbert Street came to an end in 1963 when Port Vale dished out their then-record transfer fee of £15,000 for his services after he'd made almost 140 appearances for the Foxes, chipping in with a fine 43 goals along the way. His final professional club was Mansfield Town before his out-of-football work took him to be a butcher's assistant and then on to own his own meat business. He briefly returned to the game as a scout in Lancashire for Plymouth Argyle.