Veris graduated from Chillicothe High School, where he starred in football (playing offense and defense) and basketball in 1981. Veris played football at Stanford before he was taken in the second round of the 1985 NFL draft by the New England Patriots. During his rookie season, he recorded 14 sacks—including four in the playoffs—and started in Super Bowl XX. He finished his career in 1992 with the San Francisco 49ers and later was named to the Patriots' All-1980s team
Since March 2013, Veris has worked as the Director of External Business Development & Marketing for UMass Boston Athletics.