Kaminski played junior for the Saskatoon Blades from 1984 to 1989 and was selected 48th overall in the 1987 NHL Entry Draft by the Minnesota North Stars. Kaminski only played one game for the North Stars before he was traded to the Quebec Nordiques in exchange for Gaetan Duchesne in 1989.
After spending several years in the Nordiques organization, he was traded to the Washington Capitals prior to the 1993–94 NHL season. During his four seasons with the Capitals, his hard-nosed, gritty style of play would make him a fan favorite (his nickname was "Killer Kaminski"), as he would not hesitate fighting players who were much bigger than him. As an example, on January 26, 1997, Kaminski, then playing for Washington, goaded Edmonton Oilers enforcer Louie DeBrusk into taking 27 penalty minutes just three minutes into the game, and goaded Edmonton winger Andrei Kovalenko into taking a roughing penalty before leaving the game with an injury.
After leaving the Capitals in 1997 Kaminski played three more seasons in the minors before retiring in 1999.
Kaminski was an assistant coach of the Cincinnati Mighty Ducks from 2000 to 2002. He then was the head coach of the Long Beach Ice Dogs, Missouri River Otters, and the Youngstown Steelhounds. He is currently the head coach of the Mississippi RiverKings of the Central Hockey League.