Greaves was born in Manor Park and raised in Hainault. He was scouted playing football while a schoolboy by Chelsea's Jimmy Thompson, and in 1955 was signed on as an apprentice to become one of "Drake's Ducklings" (Drake's Ducklings were named after manager Ted Drake in response to Manchester United's "Busby Babes"). He soon made an impression at youth level, scoring 51 goals in the 1955–56 season and 122 goals in the 1956–57 season under the tutelage of youth team coach Dick Foss. Greaves scored in the 1958 FA Youth Cup final, but Chelsea lost the two-legged tie 7–6 on aggregate after Wolverhampton Wanderers turned round a four-goal deficit with a 6–1 win in the second leg. He turned professional in the summer of 1957, though spent eight weeks working at a steel company to supplement his income during the summer break.