Elliot came from a family of civil servants, who held many high positions in the British Empire. His great-grandfather was governor of Madras, his grandfather Governor of Saint Helena. He was born in India, where his father was a judge.
Elliot won the competition in the armed weightlifting at the first modern Olympic Games in Athens in 1896. He was the first Scottish Olympic champion. In beidarmigen weightlifting he finished second behind the Dane Viggo Jensen. He also participated in the rings ( 4th place), hanging ( 5th place) and in athletics part. He also took part in the games held in Paris four years later, but was only able to win the eleventh place in the discus.