Brown was born on 22 August 1763 in Biggar, South Lanarkshire, the son of Richard Brown, a weaver, and Isabella Forrest. He studied theology at the University of Glasgow and became a licensed minister in 1786, serving in his home town of Biggar. He graduated as a Doctor of Divinity from the University of Edinburgh in absentia, in 1788.
In 1787 Brown crossed the Atlantic to serve as a minister in St Matthews Church in Halifax, Nova Scotia and also served as chaplin to the North British Society and British naval and army forces in the area. He left Scotland before his studies for his doctorate were complete; graduating in absentia a year later.
He stayed in North America until 1795. During this period he gathered historical information for a future publication on the history of North America. For the rest of his life this research continued but he failed to publish the materials. He attempted to compose the disputes between various factions in the parish and the region. He was particularly interested in the history of the Acadians both before and during their expulsion from Canada, and wrote sympathetically of them. He was strongly critical of Acting Governor Charles Lawrence for his role in the Expulsion of the Acadians. Brown came to the conclusion that the Explusion was largely the work of a Council dominated by Boston interests, a conclusion which anticipated the later position of John Bartlet Brebner's New England's Outpost. Brebner may have been influenced by Brown's work.
In 1791 Brown visited Philadelphia and met Benjamin Rush, who described Brown as "a man of genius learning and observation". Brown’s research appears to have included extended correspondence with the American historian Jeremy Belknap, whom he met during this same visit.
In 1792 he briefly returned to Scotland to marry his first wife before returning with her to Nova Scotia. He returned to Scotland permanently in 1795, finding employment as a minister of Lochmaben Church in Dumfriesshire. In 1799 he received a prestigious position at New Greyfriars in Edinburgh and in 1801 translated to the other half of the building, Old Greyfriars (the church at that time being physically split in two to serve two parishes).
In the same year (1801) he replaced the recently deceased Professor Hugh Blair as Professor of Rhetoric and Belle Lettres at the University of Edinburgh and served this role until his own death. He was aided in securing this position by a recommendation from Prince William Henry. Brown had travelled with the Prince on his return to Scotland in 1795.
Brown was elected a Fellow of the Royal Society of Edinburgh in 1803. He was elected Moderator of the General Assembly of the Church of Scotland in 1813 by his fellow ministers.
Brown continued his research after his return to Scotland (covering the period 1790 to 1815) but it was not published during his lifetime. He wrote several drafts of what he intended to be a comprehensive History of North America, but it was never completed and published.
He died in Carrington, Midlothian near Edinburgh on 19 February 1834, and is buried in Greyfriars Kirkyard in Edinburgh city centre.
He married Daniel (sic) Cranstoun at Harvieston near Dollar, Clackmannanshire on 10 September 1792. They had two children. He married a second time on 2 March 1805 to Mary Grant, the marriage taking place in Edinburgh. His final marriage was to a widow, Mary Pearson, on 10 March 1830 in Edinburgh.A Sermon on the Dangers and Duties of Seafaring Life (1793)
Love of Country Explained and Illustrated (1801)
The Standard of Excellence Established in the Gospel (1810)
Notice of the Life and Character of Prof Alexander Christison (1820)
The Belknap Papers
Removal of the French Inhabitants of Nova Scotia in 1755
Notes on the Acadian French