Born in a wooden shack on Pickwick Gap in Westbury, Saint Michael, Barbados, near Kensington Oval, Weekes was named by his father after English football team Everton (when Weekes told English cricketer Jim Laker this, Laker reportedly replied "It was a good thing your father wasn't a West Bromwich Albion fan.") Weekes is unaware of the source of DeCourcy, his middle name, although he believes there was a French influence in his family.

Weekes's family was poor and his father was forced to leave his family to work in the Trinidad oilfields when Weekes was eight. He did not return to Barbados for eleven years. In the absence of his father, Weekes and his sister were raised by his mother Lenore and an aunt, whom Weekes credits with his successful upbringing. Weekes attended St Leonard's Boys' School, where he later bragged that he never passed an exam (although he would later successfully study Hotel Management) and preferred to concentrate on sport. In addition to cricket, Weekes was also a keen football player, representing Barbados. As a boy Weekes assisted the groundsmen at Kensington Oval and often acted as a substitute fielder in exchange for free entry to the cricket, giving himself the opportunity to watch leading international cricketers at close range. At age 13 Weekes began playing for Westshire Cricket Club in the Barbados Cricket League (BCL). He would have preferred to have played for his local club, Pickwick, but the club only catered to white players.

Weekes left school in 1939, aged 14, and, not having a job, spent his days playing cricket and football. He later attributed much of his cricketing success to this time spent practising.[9] In 1943 Weekes enlisted in the Barbados Regiment and served as a Lance-Corporal until his discharge in 1947 and while he never saw active service, the fact he was in the military meant he was eligible to play cricket for Garrison Sports Club in the higher standard Barbados Cricket Association in addition to Westshire in the BCL.

Weekes also found time to work as a television and radio cricket commentator, known for his acerbic wit and deep knowledge of the game and began to play Dominoes and Bridge competitively, representing Barbados in regional Bridge championships.