Kemp was born of parents Eileen and Frank Kemp in St Bartholomew's Hospital, Smithfield, London, and grew up in Islington, London, in a working class family. He attended Rotherfield Junior School and Dame Alice Owen's school in Potters Bar. He was active in the Anna Scher Children's Theatre drama club, along with his brother, Martin. In 1968 he began appearing in TV and film, including a role in the 1972 film Hide & Seek, alongside Roy Dotrice.