Kapp was born in London and was educated at Dame Alice Owen's School and Christ's College, Cambridge. Whilst studying at Cambridge he had a number of caricatures published in both Granta and the Cambridge Magazine and held a one-man exhibition at the Fitzwilliam Museum in 1912. On graduation he set up his own studio and sold his caricatures to various weekly and monthly periodicals. He was gassed in the Great War as a young infantry officer and after convalescing joined the intelligence staff. Throughout the 1920s his reputation grew, such that he was commissioned to produce a series of lithographs of diplomats at the League of Nations for the British Museum and the National Portrait Gallery. In the Second War Kapp received several short commissions from the War Artists' Advisory Committee, most notably for a series called Life under London depicting people sheltering in the London Underground and in the crypt of St. Martin's-in-the-Fields during the Blitz. During 1946 and 1947 Kapp was commissioned by UNESCO to produce twenty portraits of the delegates at its first international congress in Paris. In 1961 the Whitechapel Art Gallery held a retrospective of over three hundred of his works. Further extensive exhibitions of his work were held at the Barber Institute of Fine Arts in both 1999 and 2001. Public collections include those of the Victoria and Albert Museum and the National Portrait Gallery. His first wife was the writer and political activist Yvonne Helene Mayer (1903–1999).