Boyce played for England schoolboys football team and for Essex Schoolboys at cricket. He joined West Ham as an apprentice in 1959 and made his first team debut in a Southern Floodlight Cup game against Millwall on 13 October 1959. His first Football League game was over a year later, on 22 October 1960, in a 5-2 home win against Preston North End. He made a total of 342 appearances for West Ham in all competitions, scoring 29 goals. This included 282 league appearances between 1960 and 1972, in which he scored 21 goals. He also made 22 FA Cup appearances scoring 5 goals, the most important of which was the winner in the 3-2 win over Preston North End in the 1964 FA Cup Final. He was also a member of the 1965 European Cup Winners Cup winning team on 19 May 1965. His final appearance for West Ham was 30 December 1972 in a 2-1 away defeat to Leicester City.
After retiring as a player, he became a key member of the coaching staff under John Lyall, a period which included the FA Cup Final victories in 1975 against Fulham and against Arsenal in 1980. He subsequently coached under Billy Bonds. Boyce also took charge of West Ham for one game as caretaker manager in February 1990. He was appointed West Ham's chief scout in September 1991, a position he held until 1995.