Alexander was educated at Glasgow Academy and Glasgow University where he graduated M.B., C.M. in 1883, from which he subsequently held resident posts at Manchester Royal Infirmary and Monsall Hospital. He also worked in the Wrexham Infirmary, and the Victoria Hospital, Burnley. After this he travelled in Europe and America before settling to his life-work as physician-superintendent of the Glasgow City Hospitals, a position he held for 22 years. At the time of joining up he was medical officer at the Belvidere Fever Hospital, Glasgow, and was also deputy medical officer of health and lecturer on infectious deceases in the University, being responsible for a number of important papers and reports on epidemiology, as well as being an active chairman of the Glasgow Eastern Medical Society. Alexander was described as being of a kindly disposition and had a reputation of being fearless in the discharge of his duty and careful for the well-being of those under him. He was known to be a man of many interests, and was deeply read in Scottish history. In April 1915, Alexander was appointed officer in command of the Thornhill Military Hospital, which he continued until July 1918, when he underwent an operation. He died in a nursing home in Glasgow. He was the son of the late Robert Johnston of Dumbarton and the husband of J A Johnston of Glenfield, Dumbarton.