Gordon Howard Eliot Hodgkin was born on 6 August 1932 in Hammersmith, London, the son of Eliot Hodgkin (1905–1973), a manager for the chemical company ICI and a noted amateur horticulturist, and his wife Katherine, a botanical illustrator. During the Second World War, Eliot Hodgkin was an RAF officer, rising to Wing Commander, and was assistant to Sefton Delmer in running his black propaganda campaign against Nazi Germany. His
maternal grandfather Gordon Hewart, 1st Viscount Hewart was a journalist, lawyer, Member of Parliament (MP) and Lord Chief Justice; and the scientist Thomas Hodgkin was his great-great-grandfather's older brother. Hodgkin was a cousin of the English still life painter Eliot Hodgkin (1905–1987). During the Second World War, Hodgkin was evacuated with his mother and sister to the US, where they lived on Long Island, New York. On returning, he was educated at Eton College and then at Bryanston School in Dorset. He had decided on a career in art in early childhood and ran away from school to pursue this. He studied at the Camberwell Art School and later at the Bath Academy of Art in Corsham, where Edward Piper studied drawing under him.