A forward from Bedford, Barrington played for the Norwich City youth side but was unlikely to break into the first team. Barrington had already played for Luton schoolboys and Birmingham were interested before Norwich signed him after Roy Massey spotted him playing at the age of 13 for his district side. This could be Kempston Reds who Barrington played for as a youngster.

At the end of the 1998/99 season, Norwich decided to let Belgrave go. After a trial spell with Plymouth in which he scored 4 goals in 2 games for their junior side and made an appearance for the reserves, Barrington joined the Pilgrims in early June 1999 for a minimal fee. In March 2000, Barrington went out on loan to Conference side Yeovil who had been impressed with his appearances for Plymouth's reserves. This move became permanent after Barrington was released by Plymouth on 9th May 2000.

Impressing for Yeovil, Barrington was named Conference Player of the Month for September 2000 and was called up for the FA semi-professional XI to take on the Southern League. He also interested former Yeovil manager Dave Webb who was keen to bring Belgrave to Southend. Sport First claimed on 17th December that Bristol Rovers were interested in signing Barrington. Southend returned for Belgrave on Transfer Deadline Day (22/3/01) with a £25000 bid for him, using money raised from the sale of Martin Carruthers to Scunthorpe. However Barrington rejected this opting to stay at Huish Park and attempt to take Yeovil into the Football League although this plan failed at the last hurdle when Yeovil lost 3-2 to Hereford on May 1st thus allowing Rushden & Diamonds to walk away with the Conference title.

Southend manager Dave Webb made a £40,000 bid for Barrington in June 2001. When this was rejected, Webb apparently turned his attentions elsewhere with the signing of Jason Harris from Hull. When this move failed to work out and Francis was released, Webb returned for Belgrave. Yeovil were only too willing to accept £40,000 for the striker on 14th September as Barrington was no longer a first choice striker at Yeovil. Barrington made his football league debut in Southend's 0-0 draw with Hartlepool on 15th September.

Southend manager Rob Newman announced on 28th February 2003 that Barrington's contract would not be renewed for the 2003/04 season. Conference sides Kettering and Hereford both approached Belgrave during March 2003 with respect to signing him. However he rejected their overtures. Aldershot then claimed that Belgrave was to sign for them but the sacking of Newman meant that Belgrave would stay at Southend and fight for his future under caretaker manager Stewart Robson. When new manager Steve Wignall was appointed, he quickly made his mark on the club and released Barrington at the beginning of May 2003 alongside a number of other reserve team and squad members.

He found a new club at the beginning of June 2003 when he signed for Conference side Farnborough. This lasted less than a year with Farnborough releasing Barrington in May 2004. At the beginning ofJuly 2004, Barrington signed for Lewes along with goalkeeper Jimmy Glass who famously scored to keep Carlisle in the Football League a few years earlier.

In July 2005, Barrington could be found on trial with Arlesey which became a permanent transfer in August 2005.

In July 2006, Barrington trained with Nationwide South club Bedford Town featuring in two of the Eagles' recent pre-season friendlies, which have seen Bedford beat Dunstable Town 2-0 and AFC Kempston Rovers 5-1. Bedford manager Nicky Platnauer said: "Barrington Belgrave has done well, but you would expect that with the pedigree he has got. The only unfortunate thing for him is he has got a suspension carrying over from last season. I've got to agree terms with Barrington, but that suspension puts a spanner in the works." The ban means that Belgrave would be unable to play for the first 35 days of the coming season.

In the summer of2007, Barrington signed for St Neots Town of the United Counties League.