Vasiliev was born in Leningrad (modern-day Saint Petersburg), Russia, to parents Ludmila Konstantinovna Vasilieva, a nurse, and Kim Mikhailovich Vasiliev. He graduated from the Institute for Physical Culture in Saint Petersburg. He settled in Chicago, Illinois in December 1997 but also spends time in Saint Petersburg.
Valova and Vasiliev were married from 1984–1992. He later married a Saint Petersburg resident named Valentina (divorced in 2000), with whom he has a daughter, Katia.
Vasiliev's parents decided to introduce him to skating when he was five because he had had pneumonia several times as a child and his doctor recommended an outdoor activity. As a single skater, Vasiliev won a Junior National title.
Coach Tamara Moskvina invited Vasiliev to switch to pair skating several times before he agreed, at age 18. Initially, he was physically ill-suited for the discipline and had much work to develop his muscles. He and his first partner, Larisa Selezneva, argued incessantly and split after three months. Moskvina then paired him with Elena Valova, with whom he continued to train in Leningrad (Saint Petersburg).
In 1984, Valova / Vasiliev won their first European title and then took gold at the 1984 Winter Olympics in Sarajevo. The deaths of several Soviet government officials, including one during the Olympics, cast a pall over the Soviet team and the athletes were told not to show too much joy. The pair took silver at their final event of the season, the 1984 World Championships.
In their final amateur season, Valova / Vasiliev took silver at the 1988 Winter Olympics behind Gordeeva / Grinkov but then prevailed over the reigning Olympic champions at the 1988 World Championships. After winning their third World title, Valova / Vasiliev retired from ISU competition. After performing for a year in Igor Bobrin's ice theatre, they signed a U.S. contract – the first Soviets to do so without losing their citizenship. The pair performed together in various shows and events until the end of 1997.