Born in Ufa, Soviet Union, Cherkasov first came to the tennis world's attention as an outstanding junior player. In 1987, he was ranked the World No. 3 junior player and finished runner-up in the boys' singles at the US Open (lost to David Wheaton in the final).
Cherkasov turned professional in 1988. In 1990, Cherkasov claimed his first top-level singles titles when he won the inaugural Kremlin Cup in Moscow, defeating Tim Mayotte in the final 6–2, 6–1. He also reached the quarter-finals of the 1990 Australian Open and US Open.
In June 1991 Cherkasov reached his career-high singles ranking of World No. 13. In November he successfully defended his Kremlin Cup title, saving two match points in a 7–6, 3–6, 7–6 win in the final against Jakob Hlasek.
In 1992 Cherkasov was a quarter-finalist at the French Open and won a men's singles bronze medal at the Olympic Games in Barcelona, notably rallying from 2 sets down to beat Pete Sampras in the third round.
In 1993 Cherkasov saved three match points in 3-hour, 54-minute quarter-final victory over Italy's Andrea Gaudenzi at Tel Aviv, to win 6–7, 7–6, 7–5 in what was the longest best-of-three set match in tour history.
In the end, his two victories at the Kremlin Cup in Moscow proved to be the only top-level titles of Cherkasov's career. He retired from the professional tour in 2000, having earned prize-money totalling $2,259,875.