Kellett came through the Bolton Wanderers' academy and made his debut for the team in a 1–0 loss against Leicester City on 22 April 2014, coming on as a substitute for Robert Hall for the last 20 minutes of the Championship match. It was the first time in eight years a Bolton-born player had made his debut for the club after coming through the academy. He again came on from the bench the following game, a 3–1 away win at Sheffield Wednesday, where he was praised for his attitude and performance. On 17 October, he joined League Two club Plymouth Argyle on a one-month loan, which was later extended for a further month. His first Plymouth appearance was a goalless draw at AFC Wimbledon four days after signing, and he scored his first career goal on 13 December, in a 3–2 win over Northampton Town at Sixfields Stadium. On 2 February 2015, transfer deadline day, Manchester United beat off competition from Plymouth to sign Kellett on loan for the rest of the season, with Saidy Janko going in the opposite direction for the same period. According to Bolton manager Neil Lennon, Kellett believed that Lennon was "winding him up" when he mentioned United's interest in him. He made 10 appearances for the Manchester United under-21s during the loan, including one appearance in the 1–0 Manchester Senior Cup semi-final defeat away to his parent club, Bolton.
Wigan Athletic
After returning to Bolton, Kellett signed a three-year contract with newly relegated League One club Wigan Athletic on 1 September 2015, moving for an undisclosed fee. He made his debut four days later, coming on for Sean Murray in the 79th minute of a 3–2 win at Chesterfield. Having not played for Wigan since March 2017, he joined League Two team Chesterfield on loan for the 2017–18 season in August, reuniting with former manager Gary Caldwell. He played 41 total matches for the relegated Spireites, scoring four times including in a 2–2 draw at local rivals Mansfield Town on 25 November. He was released by Wigan at the end of the 2017–18 season.
Notts County
On 31 May 2018, Kellett signed for Notts County of League Two on a two-year contract. He made his debut on 11 August in a 3–2 loss at Cambridge United, playing the last nine minutes in place of Nathan Thomas, and two weeks later scored the only goal of his 16 appearances to equalise in a 3–1 loss at Lincoln City. On 27 August 2019, he left by mutual consent after not playing for Notts County at the start of the National League season.
Non-league football
After being released by Notts County, Kellett signed for A.F.C. Fylde on a "short-term basis" in December 2019. He made just two appearances, both as a substitute and neither in the league, before signing for Alfreton Town in March 2020.
