Koceski was born in 1927, the youngest of nine children. His father, Leo Robert Koceski, Sr., died in 1938 when Koceski was 11 years old. Koceski began his football career at Canonsburg High School in Canonsburg, Pennsylvania. He played football at Canonsburg from 1943 to 1946, was selected as an All-WPIAL player in both 1945 and 1946, and led the WPIAL in scoring as a senior in 1946. Koceski scored 19 touchdowns in 1946 and was also a sprinter and long jumper for Canonsburg's track team.