Richter grew up in Flourtown, Pennsylvania, near Philadelphia, and idolized Philadelphia Flyers goaltender Bernie Parent. He attended and played for Germantown Academy in Fort Washington, Pennsylvania, and then Northwood School in Lake Placid, New York, graduating in 1985. He also played at the Wissahickon Skating Club. After playing for the United States in the World Junior Championships in 1985, Richter played for the University of Wisconsin–Madison from 1985 to 1987, and the Rangers made him the 28th overall pick in the 1985 NHL Entry Draft. He again represented the US in the 1986 World Junior Championships, as well as the World Championships and the 1988 Winter Olympics in Calgary, before making his NHL debut in the 1989 playoffs. Though he lost the one game in which he played, he was soon a regular member of the Rangers, posting 12 wins against 5 losses in his rookie season as the club's backup goaltender. Over the next two seasons, Richter split goaltending duties with the Rangers' veteran starter, John Vanbiesbrouck, and was selected to play for the U.S. in the 1991 Canada Cup tournament.