Graeme was the son of James Graeme of Braco and Catherine Stirling, the daughter of Sir William Stirling of Ardoch. His parents were both Jacobite sympathisers, but he himself played no part in the 1745 uprising. He inherited on the death of his father in 1736 and subsequently inherited also the barony of Gorthie, in Perthshire, on the death of his cousin, Mungo Graeme. The expenses inherent in his way of life necessitated the sale of Gorthie after his death.
Military career
He served in the Jacobite Scots brigade in Holland, being promoted to Captain in 1745, but was not involved in the rebellion, and was promoted to Lieutenant-Colonel in 1752. In 1758, he was appointed as a contractor supplying British troops in Germany during the Seven Years' War. In 1760/1, he raised the 105th Regiment of Foot for the British army, taking the rank of Colonel, until it was disbanded in 1763. Having been promoted to Major-General in 1762, he was appointed Colonel of the 49th Regiment of Foot from 1764 to 1768 and Colonel of the 19th Regiment of Foot. He was promoted to Lieutenant-General in 1772 and General in 1783.
Diplomat and courtier
Contrary to the report of Horace Walpole, Graeme was not sent by Lord Bute on a tour of German courts in order to report on the rival attractions of a number of Protestant princesses. However, in 1761 he was sent to Mecklenburg-Strelitz to carry George III's proposal of marriage to Princess Charlotte Sophia. Thereafter he was Secretary to the Queen between 1761 and 1764 and Comptroller of the Queen's Household between 1765 and 1774. Between 1764 and 1773, he was also Member of Parliament for Perthshire. For his services to Queen Charlotte between 1761 and 1774, Graeme was appointed, in ca. 1788, Master of St. Katherine's Hospital, a peculiar and exempt jurisdiction in the patronage and gift of the Queen Consort. This was a charitable institution, a hospice for the needy and infirm founded in the twelfth century, attached to the collegiate church of St. Katherine the Virgin and Martyr. By the eighteenth century, its main activities appear to have been supporting a school for the education of poor children and almshouses for retired sailors. It still collected income from various ancient rights, including the right of issuing marriage licences as well as grants of probate and administration. The Master was provided with a house and a modest stipend. The hospital was just outside the Tower of London, on the south-east, until 1825 when it was removed to Regent's Park and the church was demolished to make way for the construction of the eponymous dock.