Valeri was the eighth choice (129th overall) of the Quebec Nordiques in the 1988 Entry Draft. His first season in Quebec came in 1991-92, as he played in 23 games and scored 21 points (seven goals, 14 assists). In 1992-93, he notched 15 goals and 22 assists in 32 games. His playing time was limited due to injury during his first two seasons, but a healthy Valeri played 76 games in 1993-94, scoring 28 goals and 37 assists. Through 40 games in 1994-95, he scored 10 goals and 20 assists. He scored the first goal in Colorado Avalanche history on Oct. 6, 1995, setting the stage for a season of career-bests. In 1995-96, he tallied 38 goals (a team-leading 18 of which were on the power play), 47 assists, 85 points, and 85 penalty minutes. At the end of the season, Valeri and the Avalanche won their first Stanley Cup. In 1996-97, he finished the season with 28 goals and 38 assists in 68 games. In 1997-98, he recorded 26 goals and a team leading 40 assists in 75 games. From the age of 16, Valeri played in the Russian Elite league. In 1982-83, he began his career with Khimik, playing in five games. Over the next two seasons, he would score 11 goals and five assists in 65 games for Khimik. In 1985-86, Valeri went to play for the Central Red Army, where he would remain for the next six seasons. In 1987-88, he scored a career-high 26 goals and 46 points. In 1990-91, he recorded a career-best 26 assists, and tied his career-high 46 points. In all, Valeri played 259 games for the Central Red Army, scoring 111 goals and 91 assists. Valeri was a member of the gold medal-winning Russian Olympic team in 1988. He represented Russia again at the 1998 Winter Olympics in Nagano, Japan.

Alexei's NHL career began in 1993-94 with the Ottawa Senators. He was the only rookie representative at the All-Star Game that season and scored the game-winning goal. In his rookie campaign, the centerman notched 30 goals and added 49 assists in 83 games with the Senators. In 1992, Alexei was a member of the gold medal-winning team with Team Russia at the World Junior Hockey Championships. In 1993 he was a member of the Russian team in the World Championships, which also won the gold medal. In 1996, he participated in the World Cup of Hockey on Team Russia. Alexei also won a silver medal with Team Russia at the 1998 Winter Olympics in Nagano, Japan.