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Hamilton was born at Holyrood Palace, the son of the 6th Duke of Hamilton and his wife, Elizabeth. Styled as Marquess of Clydesdale from his birth until his father's death, he succeeded to his father's title of Duke of Hamilton in 1758, at the age of two.
Hamilton was educated at Eton from 1763 to 1767. However, he died in 1769, aged 14, at Hamilton Palace from a fever. He was interred in the family mausoleum at Hamilton, Scotland, and his titles passed to his younger brother, Douglas Hamilton.
- Nicolas, Sir Nicholas Harris; Courthope, William (1857). The Historic Peerage of England: Exhibiting, Under Alphabetical Arrangement, the Origin, Descent, and Present State of Every Title of Peerage which Has Existed in this Country Since the Conquest ; Being a New Edition of the "Synopsis of the Peerage of England". John Murray. p. 71.
James George Hamilton, 7th Duke of Hamilton and 4th Duke of Brandon.