Beckett began his career with Scottish junior club Renfrew.
In 1951, he signed for Rangers, making his only league appearance on 10 March 1951 in a 4–0 victory against Clyde.
Beckett subsequently joined English Kent League club Ashford Town.
In August 1957, Beckett signed for Chelmsford City, following offers from Football League clubs Halifax Town, Darlington and Crewe Alexandra.
This biographical article related to a Scottish midfielder is a stub.