Henry was educated at St Ninian's Parish School then Stirling High School. He then studied at the University of Edinburgh. After teaching at Annan, he entered the Church of Scotland, being licensed in 1746 and first preaching in Annan. In 1748, as a dissenter, he moved to Carlisle, and in 1760 to Berwick-upon-Tweed.
He then was translated to be minister at New Greyfriars in Edinburgh in 1768. The University of Edinburgh granted him a doctorate (DD) in 1771. At this time he lived at Bristo Street, just south of Greyfriars Church.
He was Moderator of the General Assembly of the Church of Scotland in 1774.
In 1776 he moved from New Greyfriars to Old Greyfriars (the other half of the same building) and remained in this role until death.
In 1783 he was one of the co-founders of the Royal Society of Edinburgh.
He died on 24 November 1790 at his home in Merchant Street in Edinburgh (just east of Greyfriars) but is buried with his family in Polmont churchyard.
He was married to Anne Balderston.
He wrote the 6-volume History of Great Britain on a New Plan (1771), covering the period from the first Roman invasion until the reign of King Henry VIII. The novelty consisted in dividing the subjects into different heads, civil history, military, social, and so on, and following out each of them separately. The work was mainly a compilation, having no critical qualities. Despite the persistent and ferocious attacks of Dr Gilbert Stewart, it was successful, and brought the author over £3000. It attracted the support of the Earl of Mansfield, whose persuasion gained for Henry a government pension of £100.