In 1958 Kemp joined the Radio Drama Company by winning the Carlton Hobbs Bursary His television credits include: Colditz, Space: 1999 and a number of other series such as: Hart to Hart, The Greatest American Hero, The Fall Guy, The Adventures of Sherlock Holmes, Conan the Adventurer, Star Trek: The Next Generation, The Winds of War, War and Remembrance and Murder, She Wrote. He played King Leontes in the BBC Television production of The Winter's Tale (1981). He also appeared as Cornwall in the 1984 TV movie version of King Lear opposite Laurence Olivier as Lear.
From the mid-1960s to the mid-1970s, Kemp had a prominent film career, usually appearing as second male leads or top supporting roles. His films include Dr. Terror's House of Horrors, Operation Crossbow, The Blue Max, Darling Lili, A Bridge Too Far, The Seven-Per-Cent Solution, Top Secret! and Four Weddings and a Funeral.
