Drummond was born at the Jubilee Hospital in Kingston, Jamaica, to Doris Monroe and Uriah Drummond. He was educated at Kingston's Alpha Boys School, where he later taught his younger schoolmate Rico Rodriguez to play the trombone.
His musical career began in 1950 with the Eric Dean's All-Stars. He continued into the 1960s with others, including Kenny Williams.
With the birth of ska Don joined The Skatalites. With Drummond's politicized conversion to the Rastafari movement, other band members followed his lead. He became a household name in Jamaica, before suffering mental problems. He was rated by pianist George Shearing to be among the world's top five trombone players.
In 1965 Drummond was convicted of the murder of his longtime girlfriend, Anita "Marguerita" Mahfood, an exotic rhumba dancer and singer, on 1 January 1965. He was imprisoned at Belle Vue Asylum, Kingston, where he remained until his death four years later. The official cause of death was "natural causes", possibly heart failure caused by malnutrition or improper medication, but other theories were put forward; some of his colleagues believed it was a government plot against the Kingston musical scene, and some believed that he was killed by gangsters as revenge for the murder of Mahfood