Dalton was born in Colwyn Bay, North Wales, to an English father, Peter Dalton Leggett, who was a captain in the Special Operations Executive during the Second World War and was an advertising executive at the time of his son's birth, and an American mother, Dorothy Scholes, of Italian and Irish descent. Before his fourth birthday, the family moved back to England to Belper in Derbyshire. While in Belper, he attended the Herbert Strutt Grammar School. As a teenager, he was a member of the Air Training Corps at LXX (Croft & Culcheth) Squadron. He decided to become an actor at 16 after seeing a production of Macbeth and got a role in a production of the play at The Old Vic. He left school in 1964 to enrol in the Royal Academy of Dramatic Art and tour with the National Youth Theatre. Dalton did not complete his RADA studies, leaving the academy in 1966 to join the ensemble of the Birmingham Repertory Theatre.