After karting, Kellett competed in the Ontario Formula Ford Championship in 2011 where he finished third. In 2012 he competed in the U.S. F2000 National Championship for Pabst Racing Services and finished 11th in the 2012 U.S. F2000 Winterfest and 14th in the 2012 U.S. F2000 National Championship season. In 2013 he returned to Pabst to race in U.S. F2000. Kellett made his Pro Mazda Championship debut in July 2013 for Team Pelfrey at Canadian Tire Motorsports Park, finishing 10th and 8th in two races. The following months Kellett and Pabst announced that they would join the Indy Lights series for the series' race at Baltimore and contest the 2014 season in the series together. Kellett crashed out on the second lap of the Baltimore race and afterwards in an interview stated that running Indy Lights with Pabst was only one of the options he was considering for 2014. On November 14 he announced that he would race in the Pro Mazda Championship with Team Pelfrey in 2014. Kellett finished 10th in the championship with one podium finish at Houston. He returned to the series in 2015 with Andretti Autosport. He again finished 10th in the championship with one podium finish, this time a second place at Iowa Speedway.
On September 22, 2015 Kellett announced that he would stay with Andretti Autosport to compete in the Indy Lights series in 2016, becoming the first driver to confirm plans for the series for the 2016 season.