Alexander Graf was the son of Leopold Graf, an Austrian professor of medicine, and his wife Marie. He was educated at the universities of Salzburg and Vienna, qualifying as Doctor of Medicine (MD) at Vienna in 1848, but at some stage also obtained the degree Doctor of Philosophy (PhD). Being implicated in the revolution of 1848 he had to flee Europe and settled in Algiers. He then moved to the Cape Colony, where he was licensed to practice in August 1852, and settled in Caledon. However, after only a few months he moved to Somerset West, where he remained for some twelve years.
In 1862 Graf was chosen as one of only ten meteorological observers outside Cape Town who were to be supplied with a set of instruments by the Cape of Good Hope Meteorological Commission. He also contributed to the natural history collections of the South African Museum. One of his donations was a beautiful little white heron, perhaps the cattle egret.
By 1866 Dr Graf was living in Cape Town, where he served as a member of the Meteorological Commission. At his death of chronic heart disease in October 1867, during a fever epidemic, he was described as one of the Cape's noblest, most talented, smart and unassuming members of the medical profession. He was married to Elizabeth Ernestina Julie Wocke and was survived by his wife, a son and four daughters.