Chapman was educated at Westminster School, Christ Church, Oxford and the London School of Economics. He served in the Royal Fusiliers in 1914–1920, particularly in France and Belgium in 1915–1918. He was appointed an Officer of the Order of the British Empire in the 1919 New Year Honours. and awarded the Military Cross in December 1919:
For conspicuous gallantry and leadership near Ghissignies, on 4th November, 1918. When the battalion was going through Gihissignies in support of the attack, information was received that the left company of the leading battalion had been held up. Under heavy shell fire he went forward to reconnoitre, and found that the supporting battalion appeared likely to become prematurely involved in the fighting. By his energy and initiative in taking command of the situation this was prevented.
He married Margaret Storm Jameson in 1926.
From 1920 to 1940, Chapman was involved in book publishing, and then served in the Second World War. From 1945 to 1953 he served as professor of Modern History at the University of Leeds, with a stint in 1948–1949 as a visiting professor at the University of Pittsburgh.
- A Passionate Prodigality, 1933
- Beckford, a biography, 1938
- A Bibliography of the Works of William Beckford, 1931
- Culture and Survival, 1940
- The Dreyfus Case: A Reassessment, 1955
- The Third Republic of France: the First Phase, 1963
- Why France Collapsed, 1968