Morrison was born near Forres, Scotland. His father, Donald Morrison, a farmer of good education who became factor to Sir Thomas Dick Lauder, married Catherine Fraser, a woman of strong Christian character, and Alexander was their sixth son. He was educated at the Elgin Academy and King's College, University of Aberdeen, where he took his M.A. degree in 1851. He was a master for two years at a school at Elgin, and then for three years was in charge of St John's Grammar School, Hamilton. During this period the number of boys at the school increased from 194 to 397. In 1856 he accepted the position of headmaster of the Scotch College, Melbourne, arrived on 26 July 1857, and a week later began his duties.
When Morrison came to Melbourne there were only 50 day boys and six boarders at the school, but in a few years it became one of the leading public schools in Australia, with a high reputation for scholarship. In 1873 considerable additions were made to the school buildings, including a house for the principal, but following a severe illness in 1874 Morrison was given a year's leave of absence and travelled widely in Europe. He was appointed a member of the council of the University of Melbourne in 1878, and for the remainder of his life was one of the most regular attendants at its meetings.
In November 1876 he moved the motion at the General Assembly of the Presbyterian Church of Victoria which led to the founding of Ormond College at the University of Melbourne, and he largely influenced Francis Ormond in his endowing of the college. He worked hard himself in obtaining subscriptions when the college was instituted, was elected chairman of the trustees, and presided at the opening ceremony on 18 March 1881. In his earlier years at Scotch College, Morrison took classes in several subjects, but as the school increased in numbers his work became largely confined to administration.
Morrison was tall, erect, black bearded, stern-looking. He was a strict but just disciplinarian who, though he mellowed as he grew older, did not quite gain the affection of his boys in the same way as Lawrence Arthur Adamson, William Still Littlejohn and Albert Bythesea Weigall. He set a high standard of scholarship in the school and never lost his grip of the conduct of it. He had the great merit of recognizing a good man when he saw him, and, at a time when there was little organization in the training of teachers, kept a high average in the quality of his staff. He trained and encouraged Frank Shew (1851-1934), who joined the staff in 1870 and for 53 years was beloved by succeeding generations of boys (see W. J. Turner's eulogy in Blow for Balloons, chapter XXVI. Turner's account of Robert Morrison, however, is a baseless travesty. Robert Morrison, a younger brother of Dr Morrison, was in fact a first-rate mathematical master, vice-principal of the school for many years, and second only to Shew in the affection of the boys). Other distinguished masters were Weigall, Alexander Sutherland, and W. F. Ingram. This was perhaps the most important factor in Morrison's 47 successful years in charge of Scotch College, but his personality was felt in other ways in the school, and his wide general interests enabled him to be an important figure in all matters relating to education in Victoria whether at the council table of the university, or when preparing and giving evidence for a Royal Commission.
He married in 1855, Christina, daughter of Donald Fraser, who died in 1883. He died suddenly on 31 May 1903. He was survived by four sons and three daughters.