During his time at Dens Park, Hamilton won 24 caps for the Scotland national team. He made his international debut in a 2–1 win over Wales in November 1961. In his 24 appearances for Scotland, Hamilton finished on the winning side on 13 occasions. His final appearance for Scotland was in a 2–1 defeat to Poland in October 1965. He also represented the Scottish League XI.[3]
Hamilton was also a noted singer and was the frontman of a band comprising several of his Dundee teammates, including future Scotland manager Craig Brown. "Hammy and the Hamsters" success was restricted to the local Dundee area however, and they only released one record.[4]
Hamilton left Dundee in 1967 to join Durban United in the South African soccer league. When Durban Spurs merged with Durban United to become Durban Spurs United, Hamilton - together with a number of other players that were surplus to the merged entity's requirements - joined the then newly promoted East London Celtic.
He played with distinction for East London Celtic which later became East London United. Always colourful and controversial, one of his more memorable games was when East London Celtic thrashed Cape Town City 6–3. Towards the end of the game, with Cape Town City players back-pedalling furiously in fear of conceding a seventh goal, Alex Hamilton - with the ball at his feet - decided to show his disdain for the opposition by sitting down on the ball. He later apologised for these actions. East London Celtic, and later as East London United, were something of a bogey side for Cape Town City. In one memorable game at the Border Rugby Union stadium in East London, despite being packed with stars such as George Eastham, Cape Town City lost once again.
Hamilton later managed junior side Dundee Violet before returning to South Africa briefly during which time he managed East London United.
Hamilton returned to Dundee in 1988 and was back at Dens Park for a spell running the club's lottery.[5] Hamilton died at his home in Broughty Ferry on 24 July 1993, at the age of 57.[2][6][7]