Born and raised in Appleton, Wisconsin, Bleier was the oldest of four children of Bob and Ellen Bleier, who ran a tavern, Bleier’s Bar, while the family of six lived above it. He had a paper route as a youth, and graduated from Xavier High School in 1964, where he starred in football and basketball. In football, Bleier was a three-time all-state selection as running back, and won all-conference honors at both linebacker and defensive back. He was a team captain in football, basketball, and track.
Bleier played college football at the University of Notre Dame in South Bend, Indiana, and graduated in 1968 with a degree in business management. During his junior season in 1966, the Fighting Irish won the national championship and he was a team captain as a senior in 1967. He was selected in the 16th round of the 1968 NFL/AFL Draft by the Pittsburgh Steelers, 417th overall.