Alexey Kuleshov was born in 1979 in Fryazino, Russia. He began playing volleyball at age 13 and was invited to join the youth team "Iskra" in 1994. He made his debut for the Odincovo team in 1996. Playing for the powerful club "Belgorod" helped develop his skills. After winning silver at the 1999 World Youth Championship, he earned a place on the Russian national team and won silver at the 2000 Sydney Olympics. Kuleshov won numerous medals for Russia including silver and bronze at subsequent Olympics and championships over his career. He was recognized individually as the best blocker and player in Russia on multiple occasions.