Burkett started his career at West Ham United, joining the senior team in July 1958.[2] He played in the FA Youth Cup Final of 1958–59 alongside Bobby Moore, Harry Cripps and Eddie Bovington,[3] and made his League debut against Fulham on 30 April 1962. Burkett made 142 League appearances for West Ham, scoring four goals. He also made 39 cup appearances, and was a member of the FA Cup winning side of 1964. He was also a member of the European Cup Winners' Cup winning team of 1965.[4]
Burkett lost his place in the West Ham team to John Charles after an injury during the 1966–67 season, and moved to Charlton Athletic for £10,000 on 12 June 1968.[5] He later played for Millwall and Southend United although he never featured in the League for either team.[6]
Burkett signed as player-manager at League of Ireland team St Patrick's Athletic in August 1971.[7] In four seasons Burkett guided Pats to an FAI Cup Final in 1974 and several high place finishes. However, he was let go in April 1975.[8]