Volkov debuted at the 1968 European Championship, finishing in 12th place. In 1974, he won the silver medal at the European and World Championships. Volkov won the World title in 1975, becoming the first Soviet men's skater to do so. He won two USSR national titles, in 1974 and 1976.
His sister is Elena Buriak, an international skating referee. With his first wife, Lyudmila, he had a son, Alexander, and with his second wife, Oksana, he had twin daughters, Ekaterina and Anastasia. He died from stomach cancer on 31 August 1990 in Kharkiv and was buried in Kuntsevo Cemetery in Moscow.