Gillan was born on 19 August 1945 at Chiswick Maternity Hospital. His father, Bill, was a storekeeper at a factory in London, who came from Govan, Glasgow and left school at 13, while his mother, Audrey, came from a family where she was the eldest of four children, who all enjoyed music and singing, and whose father had been an opera singer and amateur pianist. His sister, Pauline, was born in 1948. One of Gillan's earliest musical memories was of his mother playing "Blue Rondo à la Turk" on the piano.
He grew up moving between council flats before settling in a three-bedroom semi-detached on a council estate in Cranford, Hounslow, he was fond of animals in his early life, and enjoyed reading the comic strips of Dan Dare. His parents separated after Audrey discovered Bill had had an affair that started while he was stationed in the army during World War II. He began attending Hounslow College and stayed there through his early teenage years. He was influenced by Elvis Presley by hearing his records at home and at the local youth club. Gillan briefly attended Acton County Grammar School (now Acton High School) to take his O Levels but became distracted from studies after leaving the local cinema having watched a Presley film, deciding that he wanted to be a movie actor. He subsequently took a job manufacturing ice machines in Hounslow.