Claiborne attended Ouachita Baptist University. He was determined to attend law school at Louisiana State University so he hitchhiked to Baton Rouge, Louisiana, seeking admission. At the time one only needed two years of college for admittance to law school. After he was refused admission, Claiborne hitchhiked back home to Arkansas and caught a ride with a salesman named Schneider, who told Claiborne that he was childhood friends with Judge Albert B. Neil, the chief justice of the Tennessee Supreme Court. Mr. Schneider gave Claiborne a letter of introduction, and after visiting Neil in person, Claiborne was granted admission to Cumberland School of Law, then part of Howard College in Tennessee. He graduated from Cumberland with a Bachelor of Laws in June 1941 and returned to Arkansas, where he was a clerk in the Coulter Law Firm of Little Rock (Pulaski County).