James Douglas Hamilton Dickson was born on 1 May 1849 at 144 Bath Street, Glasgow, the son of John Robert Dickson, physician. Dickson attended Hale School. He was fifteen when he matriculated at Glasgow University, where he studied from 1864 to 1869.. At Glasgow Dickson was under the tutelage of William Thomson and assisted him with his mathematical work on electricity and deep-sea cables.

In 1870 Thomson urged Dickson to go up Peterhouse, Cambridge, to read mathematics. At the time Peterhouse was a popular college for Scottish students and also had the reputation of being mathematical. Dickson was admitted in July 1870 and was elected to a fellowship on 29 October 1974, which he held for life. He also became a senior fellow of the college in 1907. In about 1878 Dickson married Isabella Catherine Banks. They went on to have a son and a daughter.

Dickson published occasional papers, principally on thermodynamics and thermoelectricity, which appeared in the Transactions of the Royal Society of Edinburgh, Philosophical Magazine, and the Proceedings of the London Mathematical Society. He was also one of the three literary executors to whom was entrusted the publication of the collected papers of his brother-in-law Sir James Dewar. He was a fellow of the Royal Society of Edinburgh, and a member of the London Mathematical Society and of the Royal Institution.

Dickson's interests were varied. He became involved in a special study of Japan and the Japanese; contributed biographies of E. J. Routh, P. G. Tait, J. Hamblin Smith, and James Porter (all mathematicians and fellows at Peterhouse) to the Dictionary of National Biography and was also an enthusiastic musician. Dickson died on 6 February 1931 at his home in Cambridge.