Burke was born in Louisville, Kentucky and educated in parochial there, graduating from St. Xavier High School. He attended the University of Southern California and received a degree from Xavier University and a law degree from the University of Louisville. He began practicing law in Louisville in 1948. Burke served in the United States Army during World War II. Burke held appointed local offices under mayor Charles R. Farnsley including as assistant city attorney of Louisville in 1950 and 1951, director of public safety of Louisville in 1952, and executive assistant to the Mayor of Louisville, Kentucky in 1952 and 1953. Burke was elected to the Kentucky House of Representatives and served there in 1957 and 1958.

In 1958 Burke was elected to the United States House of Representatives from Kentucky's Third District (Louisville). Burke defeated incumbent Republican John M. Robsion, Jr. for the seat. Burke was re-elected to the House seat in 1960 and served from January 3, 1959 to January 3, 1963. He was defeated in a close race for re-election in 1962 by Gene Snyder; Snyder won 93,627 votes to Burke's 91,062.

In 1969, Republicans had controlled county and city offices in Louisville for eight years. Burke, a Democrat, was elected Mayor of Louisville, Kentucky that year, winning 48,337 votes to Republican John Porter Sawyer's 40,810. Burke served in that office from 1969 to 1973. Burke's priorities included reducing crime; he increased the size of the police force and installed many new street lights and the crime rate in Louisville dropped while Burke was mayor. Burke also increased garbage pickups and the sewers and pushed the construction of the Louisville Belvedere. Burke also instituted a retirement system for city employees.