Born in London, Greenham studied English at Magdalen College, Oxford before studying Fine Art at Byam Shaw. He was elected a member of the RA in 1960 and in 1964 succeeded Sir Henry Rushbury as Keeper of the Royal Academy Schools, a position he held with distinction until his retirement in 1985. He was an influential figure in the post-War generation of British artists having taught many well-known modern British artists including Martin Yeoman and Peter Kuhfeld. He served as President of the Royal Society of British Artists and was a prolific writer on art. He is perhaps best remembered as a portraitist and was commissioned by the Welsh Guards to paint H M the Queen in 1962. His work is represented in the collections of the Royal Academy and the Tate.
