He was educated at King"s College, Aberdeen from 1729 to 1733. Graduating with a Master of Arts degree on 29 March 1733.
He was ordained deacon in 1737 and presbyter at Aberdeen on 25 April 1738. He was appointed the Episcopal Incumbent of Peterhead in 1737. Along with bishops Arthur Petrie and John Skinner, Kilgour consecrated Samuel Seabury, an American Episcopal priest, as a bishop on 14 November 1784.
He resigned as Bishop of Aberdeen in 1786, and as Primus in 1788, both post were succeeded by his Coadjutor, John Skinner.
He died at Peterhead on 23 March 1790, aged 76.