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 where he performed jazz. He continued into the 1960s with others, including Kenny Williams.
After performing jazz for a decade, Drummond began performing ska and in 1964 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 ruled criminally insane and imprisoned at Bellevue 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. Heather Augustyn, author of a biography of Drummond published in 2013 claimed to have proved that Drummond's death was caused by his medications.
In 2013, a ballet telling the story of Drummond's life was performed by the National Dance Theatre Company. Created by Clive Thompson, the ballet is titled Malungu, which was Mahfood's pet name for Drummond.
In 2013 a comprehensive biography of Don Drummond was published by McFarland Publishing. Don Drummond: The Genius and Tragedy of the World's Greatest Trombonist by Heather Augustyn features a foreword by Delfeayo Marsalis.