Kelly was redshirted his first year at Clemson in 2012. As a freshman in 2013, he appeared in five games, passing for 58 yards and rushing for 117 yards with a touchdown. In April 2014, Kelly was dismissed from Clemson’s football team due to conduct detrimental to the team.

In December 2014, Kelly committed to the University of Mississippi under head coach Hugh Freeze. In his first year at Ole Miss, Kelly was named the starting quarterback.

In December 2014, Kelly was arrested following an altercation outside of a Buffalo, New York, nightclub in which he fought with bouncers and resisted arrest. Criminal charges were dropped in exchange for his plea of guilty to disorderly conduct, and he was ordered to complete fifty hours of community service.

Kelly played in his first game with Mississippi on September 5, 2015, against UT Martin and completed 9 for 15 pass attempts for 211 yards and two touchdowns while adding a 20-yard rushing touchdown. On September 19, 2015, he led Ole Miss to its second victory ever at Tuscaloosa. In the 43–37 victory over No. 2 Alabama, he completed 18 out of 33 passes for 341 passing yards and 3 passing touchdowns. On November 7, 2015, Kelly accounted for 478 total yards and six touchdowns against Arkansas. Although he completed 24-of-34 passes for 368 passing yards and three passing touchdowns, Mississippi lost to Arkansas, 52–53, in overtime. The following week, he threw two touchdown passes and rushed for two touchdowns in a 38–17 victory over No. 15 LSU. On January 1, 2016, he helped lead Ole Miss to a 48–20 victory over No. 16 Oklahoma State in the 2016 All-State Sugar Bowl. He won Sugar Bowl MVP Honors after accounting for 21 completions out of 33 passes for 302 passing yards, four touchdowns, and one interception. Kelly also led the team with 73 rushing yards on 10 carries. His 4 passing touchdowns tied a Sugar Bowl record and were most ever by a Rebels’ quarterback in the Sugar Bowl.

Kelly returned for his senior season in 2016. On November 5, 2016, Kelly suffered an injury during a 37–27 victory over Georgia Southern. The following day, it was reported that he would miss the rest of the season due to a torn ACL and a torn lateral meniscus. On February 16, 2017, the NFL rescinded an offer previously made to Kelly to participate in the NFL Scouting Combine.

Kelly was drafted by the Denver Broncos in the seventh round, 253rd overall (Mr. Irrelevant), the last selection in the 2017 NFL Draft. He was placed on the reserve/non-football injury list after having offseason wrist surgery.

Going into mini-camps for the 2018 season, he would compete with 2016 first-round draft pick Paxton Lynch for the backup quarterback role behind newly acquired Case Keenum. On August 13, 2018, Kelly was promoted to second team to act as Case Keenum’s backup. In Week 6 of the 2018 season, Kelly appeared in his first NFL game, kneeling for a one-yard loss at the end of the first half.

On October 24, 2018, the Broncos released Kelly following his arrest on suspicion of first-degree criminal trespassing. He allegedly entered a woman’s place of residence after leaving teammate Von Miller’s Halloween party. On March 20, 2019, Kelly pleaded guilty to misdemeanor second-degree criminal trespassing for the incident.

On May 20, 2019, Kelly signed with the Indianapolis Colts. He was suspended the first two games of the 2019 season for violating the NFL’s personal conduct policy. After being reinstated from suspension, he was waived on September 18, 2019. He was re-signed to the team’s practice squad the next day. He was promoted to the active roster on November 9, 2019.

On September 5, 2020, Kelly was waived by the Colts and signed to the practice squad the next day. He was released on September 30.