Kempný played as a youth in Czech Republic with HC Dyje Břeclav and made his professional debut in the Slovak Extraliga with HK 36 Skalica.

Upon competing in his first senior international tournament for the Czech Republic at the 2016 IIHF World Championships, on 23 May 2016, Kempný as an undrafted free agent was signed to a one-year contract with the Chicago Blackhawks of the NHL.

Kempný scored his first career playoff goal in Game 1 of the 2018 Eastern Conference Finals against the Tampa Bay Lightning on a wrist shot from the point.

[11] On 27 December 2021, the Capitals recalled him back to active duty after they lost six players due to the league's coronavirus protocols, including three of their top six defensemen.

[12] Unable to make the Kraken out of training camp, Kempný was placed on waivers and re-assigned to AHL affiliate, Coachella Valley Firebirds to begin the 2022–23 season.