Born in Amsterdam, Kieft started his professional career with local AFC Ajax, making his first-team debuts on 4 May 1980 at not yet 18. Amongst his youth teammates were Frank Rijkaard, John van 't Schip, Marco van Basten and Gerald Vanenburg, as in his first two full seasons he scored at an astonishing rate, especially in 1981–82 when he netted 32 goals in as many games, being crucial as the team lifted the Eredivisie title and receiving the European Golden Boot award.