Kirkup played junior football with West Ham United, and was a member of the FA Youth Cup Final team of 1956–57 alongside John Lyall and John Smith.
After signing for the senior team at the age of 17, he made his League debut against Manchester City in December 1958.
[3][4] Kirkup made 165 League appearances for West Ham, scoring six goals,[2] before joining Chelsea in March 1966 for £27,000.
He made 53 League appearances for the west London club, and joined Southampton in 1967.
He returned to England in mid-1976 and joined Byfleet Machine & Tool Company.