Kotov was born in Tula in Russia to a large working class family. He moved to Moscow in 1939 to study engineering, and also studied chess a great deal.
Kotov is best remembered today as an author. His trilogy of books Think Like a Grandmaster, Play Like a Grandmaster and Plan Like a Grandmaster are his best known, with Think Like... (which was translated into English by Bernard Cafferty and published by Batsford in 1971) being particularly famous. The book does is not concerned with advising where pieces should be placed on the board, or tactical motifs, but rather with the method of thinking that should be employed during a game. Kotov's advice to identify "candidate moves" and methodically examine them to build up an "analysis tree" remains well known today.
Kotov also had a number of good results as a player. One of his best early results was his second place finish in the 1939 USSR Championship, just missing out to Mikhail Botvinnik in the final round. This result won him the Grandmaster title, the third Soviet player to hold the title after Botvinnik and Grigory Levenfish. He won the title jointly with David Bronstein in 1948, and won at Venice in 1950, ahead of Vasily Smyslov.
In the first ever Candidiates Tournament of 1950 (the tournament to determine who challenges the World Champion, who at the time was Botvinnik) held in Budapest, he scored 8.5/18. He had qualified for the event by finishing fourth in the 1948 Interzonal Tournament in Stockholm, scoring 11.5/19.
Perhaps his best result came at the 1952 Saltsjobaden Interzonal, which he won with a score of 16.5/20, three clear points ahead of Tigran Petrosian and Mark Taimanov in second place, and without losing a game. In the following Candidates Tournament in Zürich, he scored 14/28, and was the only person to win a game against the tournament's winner, Smyslov.
Kotov played for the USSR at the Chess Olympiads of 1952 and 1954. After 1960, all the tournaments he took part in were outside the USSR. They include a shared first place with Svetozar Gligoric at Hastings in 1962, half a point ahead of Smyslov. He played in very few tournaments in his later years.