Kempe, a left wing, was chosen by the Los Angeles Kings in the first round (No. 29) in the 2014 NHL Draft.

A native of Kramfors, Sweden, Kempe played two seasons with Modo of the Swedish Hockey League before signing a three-year contract with the Kings on April 15, 2015. He was assigned to Manchester of the American Hockey League, where he helped the Monarchs win the Calder Cup with nine points (eight goals, one assist) in 17 playoff games.

Kempe made his NHL debut on Feb. 16, 2017. He scored his first NHL goal in a 4-2 victory against the Washington Capitals at Staples Center on March 11. The 2017-18 season was Kempe's first full one in the League; he finished with 37 points (16 goals, 21 assists) and was plus-11 in 81 games.

Kempe signed a four-year, $22 million contract with the Kings on July 10, 2022, and scored an NHL career-high 41 goals that season to become the first Kings player with at least 40 since Luc Robitaille had 44 in 1993-94. He scored four consecutive goals in a 6-0 win against the Pittsburgh Penguins on Feb. 11, 2023, becoming the first Kings player to score four in a game since Dustin Brown against the Minnesota Wild on April 5, 2018. Kempe was the seventh player in the past 30 years to score four straight goals, joining David Backes (Jan. 6, 2015), Ilya Kovalchuk (Nov. 11, 2005), Dany Heatley (Oct. 29, 2005), Patrik Elias (March 30, 2003), Wendel Clark (Nov. 9, 1996) and Peter Bondra (Feb. 5, 1994).

During the 2024-25 season, Kempe became the 14th player in Kings history to reach the 400-point milestone. He was also named to Sweden's roster for the 4 Nations Face-Off, bringing international experience that included four IIHF World Championships (gold in 2018, bronze in 2024). He had one goal in three games and was one of the first six players named to Team Sweden's preliminary roster for the Olympic Winter Games Milano Cortina 2026. Kempe then signed an eight-year, $85 million contract (average annual value of $10.625 million) with the Kings on Nov. 17, 2025.

Kempe's older brother, Mario, joined the Kings organization when he signed as a free agent July 5, 2019, after he had 13 points (six goals, seven assists) in 70 games for the Arizona Coyotes from 2017-19.