Greene was a Roman Catholic of Irish and Scottish ancestry, and was born in Plymouth, Devon, England. His aunt was the musical theatre actress Evie Greene. His father, Richard Abraham Greene and his mother, Kathleen Gerrard, were both actors with the Plymouth Repertory Theatre. A descendant of four generations of actors, Greene was educated at Cardinal Vaughan Memorial School in Kensington, London, and left at age 18. He started his stage career as the proverbial spear carrier in Shakespeare's Julius Caesar in 1933. A handsome young man, Greene added to his income by modelling shirts and hats.

Greene joined the Jevan Brandon Repertory Company in 1936. He won accolades in the same year for his part in Terence Rattigan's French Without Tears, which brought him to the attention of Alexander Korda and Darryl F. Zanuck. At 20, he joined 20th Century Fox as a rival to MGM's Robert Taylor. His first film for Fox was John Ford's Four Men and a Prayer. Greene was a huge success, especially with female film goers who sent him mountains of fan mail which at its peak rivalled that of Fox star Tyrone Power. One of his most notable roles was Sir Henry Baskerville in the 1939 Sherlock Holmes film The Hound of the Baskervilles. The film marked the first pairing of Basil Rathbone and Nigel Bruce as Sherlock Holmes and Dr. Watson.

Greene interrupted his acting career to serve in World War II in the 27th Lancers, where he distinguished himself. After three months, he went to Sandhurst and was commissioned. He was promoted to Captain in the 27th Lancers in May 1944. He was relieved from duty in 1942 to appear in the British propaganda films Flying Fortress and Unpublished Story. In 1943, he appeared in The Yellow Canary while on furlough. He later toured in Shaw's Arms and the Man, entertaining the troops. Greene was discharged in December 1944 and appeared in the stage plays Desert Rats and I Capture the Castle.

However, the war effectively ruined Greene's rising career. Though he did well in the popular movie Forever Amber (1947), Greene found himself cast in a series of swashbuckling roles. Having turned away from films in favour of stage and screen and having been through a divorce from Patricia Medina, to whom he was married from 1941 to 1951, Greene was cash-strapped when Yeoman Films of Great Britain approached him for the lead role in The Adventures of Robin Hood.

Greene took the role and was an immediate success. It solved his financial problems and made him a star. He had a long love affair in the 1950s with Nancy Oakes, wealthy daughter of mining tycoon Sir Harry Oakes, who had been murdered in notorious circumstances in the Bahamas in 1943.

Amongst other TV programmes, Greene was in A Man For Loving, The Doctors, The Morecambe and Wise Show, Dixon of Dock Green, Scarf Jack and Mrs Bixby and Colonel's Daughter.

Greene died of cardiac arrest in 1985.