As a freshman for Georgia, Claxton started in five of 33 games, averaging 3.9 points, 3.9 rebounds, and 1.3 blocks per game. In his sophomore season, he became a full-time starter and averaged 13 points, 8.6 rebounds, and 2.1 blocks per game, leading his team in all three categories. Claxton earned second-team All-Southeastern Conference (SEC) from league coaches. On April 19, 2019, he declared for the 2019 NBA draft. Claxton was invited to the NBA Draft Combine, which took place in May 2019, and drew praise from CBS Sports for his performance there.
On June 20, 2019, Claxton was selected with the 31st overall pick in the 2019 NBA draft by the Brooklyn Nets. On July 7, 2019, the Nets announced that they had signed Claxton. He suffered a hamstring injury in January 2020 and was assigned to their G League affiliate, the Long Island Nets. On June 24, 2020, the Nets announced that Claxton had undergone successful arthroscopic labrum repair surgery on his left shoulder and was expected to miss the remaining 2019–20 season.