A winger, Cockram joined Tottenham Hotspur as a schoolboy in 1975 and went on to make two first team appearances for the Lilywhites towards the end of the 1983–84 season.[3][4] He was released at the end of the 1984–85 season and a two-year period followed as a football nomad, which included short periods playing for Bristol Rovers and Farnborough Town and two spells in the United States.[5] A move to Isthmian League Premier Division club St Albans City during the latter part of the 1986–87 season, in a bid to return to fitness following an achilles injury,[6] proved to be the turning point in Cockram's career.[3] He was voted the Saints' Player of the Year at the end of the 1987–88 season.[7]
After a successful period training full-time with Brentford during the 1987–88 season,[6] Cockram signed a contract with the club in March 1988.[8] He made 118 appearances and scored 17 goals for the club before his release at the end of the 1990–91 season.[8] After a spell back in non-League football with Woking,[1] Cockram made a return to the Football League with Reading in October 1991 and rejoined St Albans City prior to the beginning of the 1992–93 season.[1][9] He remained at Clarence Park until the end of 1995–96 season and scored 73 goals in 211 appearances across his six seasons with the club.[9] Cockram finished his career with spells at non-League clubs Chertsey Town and Leatherhead.[3]
In 2017, Cockram established Brentford Penguins, a Down Syndrome football club.[11] A documentary film about the club, entitled Mighty Penguins, was released in 2023.[13] Fellow former footballer David Beckham, and his son Romeo, met the Mighty Penguins team in September 2023, and presented Cockram with The Sun's Who Cares Wins Unsung Hero Award.[14]
^1 appearance and 2 goals in Herts Charity Cup, 1 appearance in Wallspan Floodlit Cup
^2 appearances and 3 goals in Herts Charity Cup, 2 appearances and 2 goals in Isthmian League Cup, 2 appearances and 1 goal in Herts Senior Cup, 1 appearance in FA Trophy, 1 appearance in General Motors Acceptance Challenge Cup, 1 appearance and 2 goals in Wycombe Floodlight Cup
^4 appearances and 1 goal in FA Trophy, 3 appearances in Herts Charity Cup, 3 appearances in East Anglian Cup, 2 appearances and 1 goal in Herts Senior Cup, 2 appearances in Eastern Floodlight Cup, 1 appearance in Isthmian League Cup, 1 appearance and 1 goal in Isthmian League Full Members Cup, 1 appearance and 1 goal in Billy Minter Trophy
^ abWhite, Eric, ed. (1989). 100 Years Of Brentford. Brentford FC. pp. 398–400. ISBN0951526200.
^Croxford, Mark; Lane, David; Waterman, Greville (2011). The Big Brentford Book of the Eighties. Sunbury, Middlesex: Legends Publishing. p. 431. ISBN978-1906796716.
^Croxford, Mark; Lane, David; Waterman, Greville (2013). The Big Brentford Book Of The Nineties. Sunbury, Middlesex: Legends Publishing. p. 476. ISBN9781906796723.