Born in Streatham, Greenham trained at the Byam Shaw School of Art and then at the Royal Academy Schools from 1926 to 1929, where he was awarded a Landseer Scholarship, the Creswick Prize, the British Institute Scholarship and the Silver Medal for painting. A versatile artist he was comfortable working in most mediums and could produce figurative groups at social occasions and pastoral Suffolk landscape with equal facility. In the 1930s and ’40s Greenham painted a series of portraits of actresses and film stars, including Greta Garbo and Jessie Matthews, inspired by cinema stills taken at the time. His 1943 portrait of Martita Hunt from this series is now hanging in the National Portrait Gallery. He exhibited regularly at the Royal Academy, New English Art Club, the London Group and The Royal Scottish Academy and was an elected member of both the Royal Society of British Artists and Royal Institute of Painters in Oil. His elder brother was the painter and portraitist Peter Greenham RA.

Works are held in the collections of the National Portrait Gallery, London, Sheffield Art Galleries, Somerville Colle, Oxford, Guernsey Museums & Galleries and the Ingram Collection.