The son of William Leechman, a farmer of Dolphinton, Lanarkshire, he was educated at the parish school; the father had taken down the quarters of Robert Baillie of Jerviswood, which had been exposed after his execution (24 December 1684) on Lanark Tolbooth.
In gratitude for this service, the Baillie family helped young Leechman to go to the University of Edinburgh, where he graduated 16 April 1724.
He was moderator of a synod at Irvine in 1740, and on 7 April 1741 preached a sermon at Glasgow "on the ... character of a minister of the gospel", which was published, and passed through several editions.
The presbytery of Glasgow refused to enrol him, alleging that he had made heretical statements in a sermon published in 1743 "On the Nature, Reasonableness, and Advantages of Prayer".
The synod of Glasgow and Ayr rejected the accusation of the presbytery, and their acquittal was confirmed by the general assembly.
In 1761 he was appointed Principal of the University of Glasgow in place of Rev Neil Campbell, but for a time continued to lecture.