After graduating from Central High School in Providence, Osmanski attended the College of the Holy Cross in Worcester, Massachusetts. He played fullback for the Crusaders from 1936 to 1938. These three seasons were some of the most successful in Holy Cross' football history with the record of 23–3–3. "Bullet Bill" was named an All-American in 1938. He was named the Most Valuable Player at the College All-Star Game in 1939. His jersey number, 25, was retired by Holy Cross.
During his playing years with the Bears, Osmanski attended dental school at Northwestern University. Following his coaching years, Osmanski opened a practice in Chicago. He died in Chicago on Christmas Day, 1996.