Campbell Dodgson was the sixth son and seventh child of William Oliver Dodgson, a London stockbroker, and Lucy Elizabeth Smith, daughter of Henley Smith who owned the Priory on the Isle of Wight. He was a distant cousin of Charles Lutwidge Dodgson, better known as author Lewis Carroll. He was born at Crayford, Kent, the 7th child and 6th son of William Oliver Dodgson, a stockbroker, and his wife, Lucy Elizabeth Smith. Dodgson was a scholar at Winchester, 1880–86, and New College, Oxford University, 1886–91. He obtained a First in Greats (ancient history and philosophy) in 1890, and a Second in Theology in 1891.
Dodgson initially worked as a tutor, attempting to help his fellow Oxonian Lord Alfred Douglas. An active poet and not-so-active student, Lord Alfred had been sent down from Magdalen College in Hilary term, and the tutorship was a last-ditch attempt to assist the poet to restart his studies and take a degree. After this push failed, Dodgson was called later in 1893 to the British Museum, where he established his career as a librarian and became an art historian specializing in works on paper (1893–1932). He learnt German, 'writing German without difficulty' (DNB, 1941-50 : 215) and made many contributions to German periodicals (ibid.).
On the retirement of Sir Sidney Colvin in 1912 Dodgson was appointed Keeper of Prints and Drawings. Dodgson specialized in early modern Flemish and German prints, and published extensively on the works of Albrecht Dürer, but he also applied his expertise to works of many other schools and periods. During the First World War (1914–18) he worked in Intelligence for the War Office; his 1918 CBE was a recognition of this work (DNB, 1941-50 : 216).
He was the editor, in the 1920s, of The Print Collector’s Quarterly. He was also a contributor to the Burlington Magazine and to the Dictionary of National Biography. Dodgson married Frances Catherine Spooner, daughter of William Archibald Spooner (Warden of New College and the eponymous author of 'Spoonerisms'), in 1913 (DNB, 1941-50 : 216). Dodgson gave generously to the British Museum during his Keepership, but at the same time amassed a very large collection of over 5,000 prints which he bequeathed to the Museum. This included the first works by Pablo Picasso and Salvador Dalí to enter that collection.