Leek started his career with Northampton Town aged 17 despite only playing football for three years. After six years, he earned a move to First Division Leicester City. He had three seasons at Leicester, and scored in every round of the FA Cup en route to the 1961 final. He was dropped from the final and soon moved on to Newcastle United. His stay with Newcastle was brief and he signed for Birmingham City before the end of the year, having had a loan spell in Canada with Montreal Concordia. He played more than 100 games with Birmingham, scoring nearly a goal every other game including two in the 1963 League Cup triumph against rivals Aston Villa. He returned to Northampton, helping them to promotion to the top flight before finishing his professional career with Bradford City. His career took him onto three Welsh non-league sides.