Alexander was a son of pastoralist George Melrose (1806–1894) and his wife Euphemia née Thomson (c. 1829–1887), who arrived in South Australia from Scotland in the Palmyra in 1839, who founded the "Rosebank" property in Mount Pleasant.
Alex was born in Australia and educated at Prince Alfred College, and studied law at the University of Adelaide.
In 1928 he was deputised by the board to travel to Great Britain and purchase three paintings to fill gaps in the gallery's collection.
He was a member of the Botanic Garden Board, was president of the Royal Institution for the Blind from 1930 and vice-president of the Pioneers' Association from 1937.
[4] Alex never married, and lived for many years at "Chiverton", Wattle Park, cared for by his niece, Alice Effie Ferguson ( –1949).