Brodhead attended Duke University where he played quarterback on the football team. During that time he led the Blue Devils to the Orange Bowl and shared the quarterback position with Sonny Jurgenson.
He was drafted by the Cleveland Browns in 1958 but went on active duty in the Armed Services before being able to join the Browns. After his discharge in 1960 he signed instead with the upstart Buffalo Bills of the fledgling American Football League. He played one season of professional football in the American Football League during the 1960 AFL season for the Buffalo Bills, starting one game before having a brief stint with the CFL Edmonton Eskimos. In 1961 he embarked on a legendary career in minor league football with the Canton Bulldogs and Cleveland Bulldogs in the United Football League and then, in 1965 and 66 with the Philadelphia Bulldogs of the Continental Football League. He led the Bulldogs to the CFL title in 1966 with a win over the Orlando Panthers, and then, at age 28, was named Business Manager of the Cleveland Browns.
Bob Brodhead was elected to the Minor League Football Hall of Fame for his career with the Canton/Cleveland/Philadelphia Bulldogs.
Brodhead remained with the Browns until 1970, when he was named General Manager of the Houston Oilers. He later became the CFO of the Miami Dolphins before heading to LSU as AD in 1981. His tenure at LSU was highly successful on the field and controversial off the field. He later was a radio personality in Baton Rouge and for a time the AD at Southeastern Louisiana University.