Murison was born in 1700, the son of James Murison, minister at Garvock.
In 1721, he obtained his M.A. from King's College, Aberdeen, and in 1729 he was presented as Minister to the parish of Edzell. In 1743, he transferred to the parish of Kinnell, before, in 1747, being appointed Principal of St Mary's College at the University of St Andrews.
Boswell and Johnson visited Principal Murison (as he then were) at St Mary's College at St Andrews on August 19, 1773, on the first leg of their tour of Scotland. Boswell wrote of the occassion that "[the college has] a good library-room; but the principal [Murison] was abundantly vain of it, for he seriously said to Dr Johnson, 'you have not such a one in England'."