Claiborne was born in 1917 in McRae, Arkansas. He attended Ouachita Baptist University and Cumberland School of Law where he received a Bachelor of Laws before serving in the United States Army during World War II. After the war he settled in Las Vegas, Nevada, where he established himself as a flamboyant and well-known defense attorney representing many prominent people with ties to Las Vegas, including entertainers like Frank Sinatra, Dean Martin, Judy Garland, and Carol Burnett, and mobsters like Bugsy Siegel, Frank "Lefty" Rosenthal, Joe Conforte, and Benny Binion.