 After graduating from Young Harris College in 1949, Brinkley became a school teacher for several years before serving as a pilot in the United States Air Force from 1951–1956.After his military service, Brinkley entered the University of Georgia School of Law in Athens, graduated with a Juris Doctor degree in 1959, and became a practicing lawyer in Columbus, Georgia.Brinkley's political career consisted of one term in the Georgia House of Representatives from 1965–1966 and eight consecutive terms in the U.S. House of Representatives beginning in 1967 and ending in 1983 when he chose not to run for re-election.