Kelly was born in Huntingdon, Cambridgeshire, and was attached to Cambridge United as a youth. He joined the Norwich City Academy in July 2010. He was released in March 2012 and joined the Everton Academy, before he left Everton in June 2013. He had a trial at Rangersin September 2013, but manager Ally McCoist was unable to offer him a contract due to financial restrictions. He joined Cambridge City of the Southern League Premier Division in February 2014. He returned to play for the Norwich under-21 side in the 2014–15 season.
Port Vale
Kelly signed a two-year contract with League One side Port Vale in June 2015. He made his debut in the Football League on 8 August in a 0–0 draw with rivals Crewe Alexandra at Gresty Road. After five further appearances he went three months without playing a game as Enoch Andoh, Sam Foley and Colin Daniel were preferred on the left-wing, before marking his return to the first team with a goal and an assist in a 3–2 win over Chesterfield at Vale Park on 21 November. For his performance he was named on the Football League Paper's team of the week. He got the chance to establish himself in the first team following a long-term injury to Andoh, and though initially inconsistent he managed to impress manager Rob Page who saw "glimpses" of his "terrific talent" by beating three defenders to score the only goal of the game against Bury on 28 December. He came off the bench against Coventry City on 7 February and scored another "sensational" goal to secure a 1–1 draw; this goal won him the club's Goal of the Season award. Kelly fell out of the first team picture under new manager Bruno Ribeiro and had to wait until the end of October for his first league start of the 2016–17 season. He then went on a run of seven consecutive starts. He was released by new manager Michael Brown following the club's relegation in May 2017.
Grimsby Town
On 31 May 2017, Kelly signed a two-year contract with League Two side Grimsby Town on a free transfer. He made 12 appearances throughout the 2017–18 campaign, of which only one came after Michael Jolley replaced Russell Slade as manager, as the "Mariners" posted an 18th-place finish. On 21 June 2018, Kelly left Blundell Park after his contract was cancelled by mutual consent.
Hamilton Academical
On 25 June 2018, Kelly signed for Scottish Premiership club Hamilton Academical. He said that he was convinced to join the club after listening to manager Martin Canning's passion and ambition. However he made only nine appearances for the "Accies" in the first half of the 2018–19 season before leaving New Douglas Park in January 2019.
Non-league
On 7 February 2019, Kelly joined National League side Braintree Town. He scored one goal in five matches for the "Iron" before he moved on to National League South side Billericay Town the following month. He ended the 2018–19 campaign with three appearances for "Ricay". On 1 June 2019, he signed with newly-promoted National League North side King's Lynn Town; "Linnets" manager Ian Culverhouse said Kelly was "a hell of a signing for this football club". He struggled for form at the start of the 2019–20 season, before improving enough to provide the assists for both of Town's goals in a 2–1 victory at Southport on 16 November. King's Lynn went on to win promotion as champions of the National League North after the league table was decided on points per game due to the COVID-19 pandemic in England.
