Craxton studied piano at the Tobias Matthay Pianoforte School and made a name for himself early in his career as an accompanist with performers such as Dame Nellie Melba, Dame Clara Butt, Lionel Tertis and John McCormack.
In 1919 he became a professor at the Royal Academy of Music. He remained there until 1961, although he continued teaching from his studio long into his later years. Some notable students included: Ronald Kinloch Anderson, Winifred Atwell, John Bingham, Joyce Howard Barrell, Howard Brown, Christine Croshaw, Elaine Hugh-Jones, Alexander Kelly, Denis Matthews, Noel Mewton-Wood, A A Owen, and Alan Richardson.
Craxton was also an active composer. His first published work was the "Three Pieces for Pianoforte" (1911). Some of his songs were recorded by John McCormack and Lauritz Melchior. He also collected musical compositions in association with Alfred Edward Moffat.
Craxton died aged 85 in 1971.