Boris Aleksandrov played only one full season in 1976 with the Soviet national hockey team, playing in 19 games, scoring four goals and winning an Olympic gold and a bronze in the Canada Cup in that season. But he had a very long career at the club level, which lasted well into the 1990s. Aleksandrov began to play hockey with Torpedo Ust-Kamenogorsk (now Oskemen) in 1972, but in 1973 joined CSKA Moskva and played there until 1978. With CSKA Moskva Aleksandrov won three Soviet Championships titles (1975, 1977, 1978) and three European Champions Cup titles (1974, 1976, 1978).
After leaving CSKA Moskva Aleksandrov played one season with SKA Moskva Oblast, before joining Spartak Moskva from 1980-82. In 1980 while playing with Spartak Moskva Aleksandrov was selected as part of the best line at the Soviet Championships. From 1982-88 Aleksandrov again played with Torpedo Ust-Kamenogorsk, before spending three seasons abroad, playing with HC Milano (Italy), Ferencváros TC (Hungary) and Alisa Moskva (Russia). Aleksandrov ended his playing career with Torpedo Oskemen from 1992-96. In 1995 he also played four games for the Kazakhstan national team and scored two goals.
From 1996 until his untimely death in 2002, Aleksandrov worked as a head coach of both Torpedo Oskemen and the Kazakhstan national hockey team. On 31 July 2002 Aleksandrov died in a head-on collision with another car en route from Chelyabinsk to Moscow.