Joseph, a forward from Laval, Quebec, was selected by the Tampa Bay Lightning in the fourth round (No. 120) of the 2015 NHL Draft after he scored 21 goals in his first full season with Saint John's of the Quebec Major Junior Hockey League.

He played two more seasons with Saint John's, scoring 33 and 36 goals, before turning pro and joining Syracuse, the Lightning's American Hockey League affiliate, for the 2017-18 season. After scoring 15 goals and 53 points in 70 games with Syracuse, Joseph earned a spot on Tampa Bay's roster for the 2018-19 season. He earned his first NHL point, an assist, on Oct. 16, 2018, in a 4-2 win against the Carolina Hurricanes and scored his first goal on Nov. 4 in a 4-3 overtime win against the Ottawa Senators.

Joseph finished his rookie season with 26 points (13 goals, 13 assists) in 70 games, helping the Lightning tie the NHL single-season record with 62 victories, equaling the mark set by the Detroit Red Wings in 1995-96. He scored seven points (four goals, three assists) in 37 games for the Lightning in 2019-20 but did not see action during Tampa Bay's run to the Stanley Cup.

Joseph was traded to the St. Louis Blues by the Ottawa Senators on July 2, 2024, with a third-round pick in the 2025 NHL Draft, for future considerations. He signed a one-year contract with the Los Angeles Kings on March 6, 2026.