He was the son of Alexander Murray, 4th Lord Elibank, and his wife Elizabeth, daughter of George Stirling of Keir, and an eminent surgeon in Edinburgh. General James Murray was his younger brother; as was Alexander Murray, who gained some notoriety as a Jacobite, not least during the 1750 by-election in Westminster. He was uncle of Major Patrick Ferguson killed at the Battle of King's Mountain in 1780. Although admitted a member of the Faculty of Advocates in 1722, he soon turned from legal to military pursuits, becoming an ensign in the army, and subsequently major in Ponsonby's foot and lieutenant-colonel in Wynyard's marines. With the latter regiment he served at the Battle of Cartagena de Indias in 1740. After the failure of that expedition Murray quit the army. He had married in 1735, and had succeeded his father as Lord Elibank the next year. Returning to Scotland, he associated chiefly with the members of the legal profession, among whom he had been brought up, and seems to have been very popular ; but his chief interests were literary. He was long in intimate relations with Lord Kames and David Hume, and the three were regarded in Edinburgh as a committee of taste in literary matters, from whose judgment there was no appeal. He was the early patron of Dr. Robertson, and of John Home the tragic poet, both of whom were at one time ministers of country parishes near his seat in East Lothian. Upon the accession of George III, Elibank, like many other Jacobites, rallied to the house of Hanover ; and when Lord Bute came into power it was determined to bring him into the House of Lords. This plan was, however, foiled by a severely sarcastic article by Wilkes in the North Briton on his presumed services to the Pretender. Wilkes had been an unsuccessful candidate for the governorship of Canada when that office was conferred on Elibank's brother, General James Murray. When in Scotland in 1773, Dr. Johnson paid Elibank a visit at his house ofBallencrieff Castle, Haddingtonshire. He was member of the Cocoa Tree Club and The Poker Club. His Edinburgh address in his final years is given as James Court on the Lawnmarket. Elibank died at Ballencrieff on 3 August 1778.