Holmes attended Roosevelt High in Washington, D.C. He was a three year lettermen and captain in track, basketball, and football at Roosevelt where he broke the school and city's record in the high jump with a 6 feet, 3 inches mark and was later named to the All-City football and track team his junior and senior years. Also, he was voted the Best Athlete in the school. Holmes was a three-year veteran of the Marine Corps. Norman also played basketball and high jumped 6 feet, 5 inches in the 1963 Junior AAU meet. He ran the quarter mile in 48.5 and the 100 yard dash in 9.8.
Holmes elected to play basketball at WVU where he was known for his efficiency. As a freshman, he shot 52.5 percent and averaged 11.8 points per game. Going into his sophomore year he was considered to be a quick, aggressive, and fine athlete. Although it was believed that his outside shooting had to be improved for him to be prime contender for a position in the Mountaineer backcourt. That year, he shot 51.2 percent while averaging 2.7 points per game. Heading into his junior year he was rated as one of the best defensive men and a good rebounder for his size. That year, Dave Palmer's career ending injury opened the door for him. He then went on to pick up his outside shooting in the closing stretch of the 1967 season. Evidenced by his 18 points against George Washington, all in the first half, his collegiate high.