Kennedy was born in Osceola, Arkansas. Lived and grew up in Wilson, Arkansas. He graduated from Rivercrest High School in Wilson, Arkansas, and attended Northwest Mississippi Community College before winning a football scholarship to the University of Miami, where he was named All-American at his position in 1989.
Kennedy was inducted into the University of Miami Sports Hall of Fame in 2004.
He has a jersey of his number in the Rivercrest High School lobby.
He was the third overall selection in the 1990 draft by the Seahawks, and was unsigned until two days before the beginning of the season. Kennedy was named to the Pro Bowl in 1991. In 1992, having recorded 14 quarterback sacks, he received the NFL Defensive Player of the Year by the Associated Press despite the Seahawks miserable 2-14 record. He switched his jersey number to 99 that season in honor of close friend Jerome Brown, and was named All-Pro 4 times.
Kennedy retired after the 2000 season. In 167 games with Seattle, he recorded 668 tackles, 58 sacks, and 3 interceptions. He announced his retirement in August 2002 after sitting out the 2001 season. He was given several offers by other teams, but wanted to finish his career in Seattle. He is generally considered one of the best defensive tackles to ever play the position in the NFL. He was a Semi-Finalist for the Pro Football Hall of Fame in 2008, as well as a finalist in 2009 and 2011, eventually being elected to the Hall as a member of the 2012 induction class.