Finch was born in Caterham, Surrey, the son of a merchant banker. He first started acting at school, later gaining experience in amateur theatre groups. After serving in a parachute regiment during his military service, he joined an SAS reserve regiment.
Finch had started acting professionally with several different repertory companies around the UK before he got a part in Crossroads, the popular daytime soap, during its first run in 1964. Finch then appeared in Z-Cars (1967-68) and in 10 episode