Leslie was commissioned lieutenant on 23 June 1755, and from April 1763 through to 1765 had the cutter Morning Star 4, serving in the North Sea and initially operating out of Hull. From 1769-71 he had another cutter, the Meredith 6, in which he was employed in the Channel.
He was promoted commander on 4 March 1779 and appointed to the Minorca 18. By January 1780 he was in command of the frigate Enterprise 28, serving in the Mediterranean. She was present at Gibraltar on 7 June when the Spanish launched a fireship attack on the shipping at the Rock, and he remained as the senior naval officer there when Captain John Harvey returned to England a few weeks afterwards. The Enterprise was later in escort of a convoy when she witnessed, but did not endeavour to take part in, the action between the British frigates Flora 36 and Crescent 28, and the Dutch frigates Briel 36 and Castor 36 in the Straits of Gibraltar on 30 May 1781. After returning to England the Enterprise was paid off in August.
In September 1781 Leslie assumed command of the Preston 50 in place of Captain Alexander Graeme who had lost an arm at the Battle of the Doggersbank, initially serving on convoy duty in home waters before going out to Jamaica in June 1782. During the early summer of 1783 he moved into the Torbay 74 which he brought home from Jamaica.
Shortly after returning to England Captain Leslie died on 19 October 1783 in the York Buildings, Buckingham Street, London.