Kholmov learned the game at the age of 12 and by 14 he was the champion of Archangelsk. From 1960 to 1970 he was one of the strongest GMs in the Soviet Union, qualifying for the Soviet Championship seventeen times.
After moving to Moscow in 1967 his state pension amounted to $50 a month which wasn't enough to live on, so he supplemented his income playing chess in the clubs around Moscow. In an interview near the end of his life Kholmov stated he didn't care for the new, faster, FIDE time controls, but felt they benefited the older players because they didn't have to play for long hours which was very tiring. On the other hand, he believed rapid controls were bad because they spoiled the game. His best solution was to let players choose between what type of tournament they wanted to play in, rapid or long games.
During his career Kholmov won many international tournaments in Eastern Europe. Kholmov made his international debut at Bucharest 1954 with a tied 3rd–4th place; Viktor Korchnoi won. Probably the best result of his career was when he tied for 1st–2nd with Smyslov at the Moscow International 1960. As a result he was awarded the GM title that same year. He tied for the Soviet Championship in 1963, but lost the playoff between him, Boris Spassky and Leonid Stein.
His biggest disappointment was the the Soviet Zonal tournament in Moscow in 1964, where he scored 6 -6 and finished in 4th place. Only the top three qualified for advancement to the Interzonal. His single appearance on the USSR team was when he played board ten at the European Team Championships in 1970 where he won the board gold medal with a score of +3 =3 −0.
In 2000 Kholmov tied for the World Senior Championship with Mark Taimanov, Janis Klovans and Alexander Chernikov. Then he placed 2nd–4th in the same event in 2001, tying with Klovans and Vladimir Karasev.
Kholmov played competitive chess virtually right up until his death in early 2006 at age 80. He appeared in a Senior event in Dresden, 50 years after he won a tournament there.
Kholmov came from a family that was politically suspect and because of that and other black marks on his record, for political reasons he was allowed to participate only in a few tournaments abroad, all of them in Socialist countries; he never got to play in a Western country until 1990 when he was 65 years old.
His father defected to the Communists during the Russian Civil War and joined the party in 1918 and then served in various government positions. In the late 1920s he was arrested and shipped off with other convicts to build the White Sea-Baltic Canal. Although many prisoners died horrible deaths, he managed to survive and made it home where he obtained another government job. But then in 1938 he was arrested again during the Great Purge, a political campaign from 1936 to 1938 which involved a large-scale purge of the Communist Party and government officials, repression of peasants and the Red Army leadership, and widespread police surveillance, suspicion of saboteurs, imprisonment, and executions. Nobody knows what happened to him...he was never heard from again.
Kholmov was a sailor in the merchant marines and spent World War Two in a Japanese prison camp after his ship was seized.