Burleigh was born in 1885 in Wyoming, NY. He attended the Klindworth-Scharwenka Conservatory in Berlin from 1903–1905, where he studied violin with Anton Witek and composition with Hugo Leichtentritt. Returning to the U.S., he attended the Chicago Music College studying composition with Felix Borowski. He enjoyed a brief career as a concert violinist from 1907–1909 then taught at a series of colleges before going to New York in 1919 to study violin with Leopold Auer and composition with Ernst Bloch. From 1921–1955 he taught violin, theory and composition at the University of Wisconsin-Madison. He died in Madison in 1980.