Henry Thurburn was the son of a merchant and was educated at the Edinburgh New Academy and Stanmore School, Middlesex. He joined the English East India Company in June 1842 and in April 1844 was posted to the 42nd Madras Native Infantry in India. In 1847 he saw active service on an expedition to China under Major-General G.C. D'Aguilar and in February 1850 was granted leave to visit Europe. In 1858, during the so-called Indian Mutiny, he was appointed second-in-command of the First Regiment of Beaton's Horse. He left India on sick leave at the end of 1859, returned to Scotland, and retired in November 1861.
By 1869 Thurburn was living in the Cape Colony. As a trustee of the marriage settlement of one John Paterson he was involved in a court action against the trustees of Paterson's insolvent estate, to establish the prior claim of his marriage settlement over the claims of his creditors. The action failed in the Cape Supreme Court in March 1869, but on appeal to the Privy Council in England the decision was reversed in January 1871. He was a trustee or liquidator also of the Umzinto Plantation and Trading Company of Natal during 1873-1875.
Thurburn collected some prehistoric artefacts from the sand dunes at Port Beaufort (at the mouth of the Bre?rivier) and at Hout Bay, near Cape Town. These were presented to the British Museum (Natural History) in January 1871 and accessed as Christy No. 7581-7600. In 1873 he collected pottery and circular stones at Port Beaufort. These were presented to the museum by John Evans in May that year and accessed as Christy No. 7805-7813.
Thurburn returned to Scotland and built a house called "Craigness" overlooking the sea at Muchalls, south of Aberdeen, where he lived until his death. He was married and had at least two daughters.