Robertson was educated at the free school at Appleby, and on 17 March 1746 matriculated from The Queen's College, Oxford.
He became rector of Sutton, Essex, in 1770, and in 1779 vicar of St. Mary's Church, Horncastle, by the gift of his relative Edmund Law.
His wife, a daughter of Timothy Raikes, chemist, of London, survived him, but his children all died in infancy.
In 1772 he revised for the press Gregory Sharpe's posthumous sermons, and in the same year edited Algernon Sidney's Discourses on Government, at the request of Thomas Hollis.
He also wrote in the Gentleman's Magazine, and produced a learned work on the authenticity of the Parian Chronicle (London, 1788), which was answered by John Hewlett.