Griff was educated at Brentwood School, Essex, where he met Charlie Bean (Executive Director of the Bank of England) and Douglas Adams, the writer of The Hitchhiker's Guide to the Galaxy. He was part of a group called The Clique by the Headmaster: they would hang around making each other laugh, and Griff has just continued doing this for the rest of his life. Rhys Jones and Bean followed Adams to the University of Cambridge.
Since then, he has appeared in numerous comedy programmes, usually with Mel Smith, including the second series of notable 1980s Britcom Not the Nine O'Clock News, alongside Smith, Rowan Atkinson and Pamela Stephenson.