Drachkovitch was born in Belgrade, Kingdom of Serbs, Croats and Slovenes (later Yugoslavia), as the posthumous son of Milorad Drašković, the country's Minister of the Interior, who had been assassinated.
A liberal Serbian nationalist during World War II, Drachkovitch initially joined Chetnik resistance to the Nazi occupation, and later fled the country to escape Communist rule.
Drachkovitch resumed his studies at the University of Geneva, receiving a bachelor's degree in political science in 1949 followed by a PhD in 1953.
After serving as the director of studies at the College of Europe in Bruges, Belgium, Drachkovitch went to the United States in 1958, where he held a succession of academic posts.
He retired from the position in 1988 but and was awarded the title of senior fellow, emeritus, at the Hoover Institution in 1993.