Born in Irvine, Ayrshire, Roy grew up in Ardrossan. He signed for Celtic as a youngster at 16 and came through their youth ranks. Aitken was nicknamed "The Bear" by their supporters due to his large frame and commanding presence. Aitken broke into the Celtic first team as a teenager, first captaining the team in October 1977. Playing as a centre back or defensive midfielder, Aitken made the third-most appearances in the club's history and was particularly noted for his contributions in the club's double-winning 1987–88 season.