Wilkinson was born in New York City, New York to J. Harvie Wilkinson Jr. and his wife. He was raised in Richmond, Virginia, where he attended St. Christopher's School during the state's Massive Resistance crisis concerning desegregation of the public schools. His father (CEO of State Planters Bank, later part of Crestar Bank) joined with Norfolk and Western Railroad CEO Stuart Saunders and Richmond School Board President (and later Supreme Court Justice) Lewis F. Powell and others to support Governor J. Lindsay Almond when he decided to break with the Byrd Organization and adhere to the decisions of the Virginia Supreme Court and a three judge federal panel on January 19, 1959, which declared certain new laws designed to maintain segregation unconstitutional.