He was educated at Magdalen College, Oxford, but left the university early without a degree, in order to assist a relative, Alderman Joseph Robinson, grocer, in business at Stamford, Lincolnshire.
In 1803 he was elected a Fellow of the Society of Antiquaries of London; and in the following year he married Elizabeth, daughter of James Nowlan, a merchant, of the Hermitage, Wapping.
He printed in 1805, for private circulation, a short volume of Rhymes; and in 1807 he published an edition of the Poems of Richard Corbet.
Notes of Gilchrist are incorporated in the third edition, by John Payne Collier) of Robert Dodsley's Old Plays, 1825–7.
The Quarterly Review for June 1812 contains a severe article by Gilchrist on Stephen Jones's edition of David Erskine Baker's Biographia Dramatica.