Born in Greymouth on 3 February 1936, Finlay was the son of Jack "Doc" Finlay.[1][2]
Trained by his father, Graham Finlay won four New Zealand national amateur boxing championship titles: the light welterweight title in 1954; and the welterweight title in consecutive years from 1955 to 1957.[2] He represented New Zealand in the welterweight division at the 1956 Olympics in Melbourne, losing to the eventual bronze medallist, Kevin Hogarth from Australia, on points in his first bout.[2] In the light-welterweight division at the 1958 British Empire and Commonwealth Games in Cardiff, Finlay was eliminated in the first round, defeated by G.R. Smith from Southern Rhodesia.[2] Overall, Finlay's amateur record was 98 fights for seven defeats.[3]
Finlay was also a useful rugby leaguesecond rower, playing for the Greymouth Marist club side that won several West Coast titles, as well as the Thacker Shield in 1959 and 1961.[2]
Outside of sport, Finlay operated the Chevron restaurant and milk bar in Greymouth with his brother-in-law, John Morris.[2] In 1985, Finlay relocated to Christchurch, where he worked for the Inland Revenue Department until his retirement.