Alexejew started between 1987 and 2001 in seven world championships, first for the Soviet Union, later for Russia. With the relay he missed a possible medal several times. He was fourth in the season three times alone. His best individual result at a world championship was a fourth place in the long distance race in 1991, when he lagged behind the Swedes Jörgen Mårtensson and Kent Olsson and the Soviet Estonian Sixten Sild .
With the Finnish team Ikaalisten Nouseva Voima he took part in the Jukola relay race several times .
