Leslie's father was Rector successively of the schools of Haddington and Dalkeith.
At some point in the late 1740s, Leslie was hired as Tutor to Sholto Douglas (1732-1774), Lord Aberdour, son of James Douglas, 16th Earl of Morton (1702-1768). In 1751 he accompanied the young Lord to Leyden University, where they stayed until 1753. Clearly satisfied with the services done by Leslie to his son, Lord Morton settled on Leslie an annuity of £40.
The next year, in 1754, he was offered the position of Professor of Greek at King's College, Aberdeen, a position which he accepted and stayed in until his death in 1790.
He died on May 24, 1790. An obituary in the Aberdeen Journal read as follows:
His attachment to his pupils, and his unwearied exertions to instruct them in the principles of languages to preserve order and regularity in their behaviour and to enforce the practice of every moral and religious duty will render his memory dear to all who have been under his care.
Unlike many of his contemporaries, Leslie did not publish any writings, and he is largely unknown today. The main source for information on his life is the 17 page biography An Aberdeen Professor of the Eighteenth Century by J. G. Burnett, published in The Scottish Historical Review, Vol. 13, no. 49 (Oct. 1915).
On August 23, 1773, in Aberdeen, Prof. Leslie, together with Dr. Gerard and Prof. MacLeod, came to meet Boswell and Dr Johnson at Sir Alexander Gordon's. Johnson had just been presented with the Freedom of the City of Aberdeen earlier in the day. Boswell later wrote that "We had little or no conversation in the morning. Now we were but barren. The professors seemed afraid to speak."