William Graham Claytor, son of Graham and Marion Bell Claytor, was born in Bedford, Virginia on December 20, 1886. More familiarly known by his middle name, Graham Claytor matriculated at Virginia Agricultural and Mechanical College (later Virginia Tech) as an electrical engineering student in 1902. Following his father's death, Claytor left school in his junior year to work as a student engineer in General Electric's Lynn, Massachusetts testing plant. In 1907, Claytor accepted a position with the Roanoke Railway and Electric Company as a salesman; in 1908, he was appointed an electrical engineer. That same year, Claytor married Gertrude Harris Boatwright, a Virginia poet. The couple would have three sons.

Through successive promotions, Claytor had become general superintendent of Roanoke Railway and Electric and the Lynchburg Traction and Light Company by 1914. His career on temporary hold while serving as a captain in the U.S. Army Quartermaster Corps from 1918 to 1919, Claytor afterward resumed his previous position and in 1923, was named chief engineer for American Electric Power Company of Philadelphia. In 1926, Claytor became the company's general manager. The following year, the company was purchased by American Gas and Electric Company of New York. Claytor was transferred to New York City and appointed vice president and director of a number of the company's subsidiaries, including Appalachian Electric Power Company, Kentucky and West Virginia Power Company, West Virginia Power Company, Radford Limestone Corporation, and Franklin Real Estate Company. As vice president of Appalachian Power, Claytor directed construction of the Appalachian Power Company's dam on the New River in Pulaski County, Virginia. Completed in 1939, the dam created Claytor Lake, named in his honor. Claytor retired in 1954, but returned to serve as executive vice president of American Electric Power from 1960. After retiring from American Electric again in 1962, he continued to serve as president of Richardson-Wayland Electric Corporation (Roanoke, Virginia).

Claytor remained an ardent supporter of his alma mater and in the 1950s served on the board of directors of both the VPI Alumni Association and the VPI Educational Foundation. Chief among Claytor's other interests was ham radio, a hobby he pursued for more than 25 years. Following his 1962 retirement, Claytor resided in the cottage he had built on Claytor Lake. Graham Claytor died in Roanoke February 28, 1971, and was buried in Evergreen Burial Park, Roanoke, Virginia.