Klee, who comes from Wattwil in Toggenburg and competes for the Speer Ebnat-Kappel ski club, took part in the U18 and U20 races in the Alpine Cup from 2013 to 2016 . He finished fourth in the U20 overall ranking in the 2015/16 season. At the European Youth Olympic Winter Festival 2013 in Predeal , he came 62nd place in sprint, 25th place over 7.5 km classic and 21st place over 10 km freestyle. At the Nordic Junior World Ski Championships 2014 in Fleimstal he ran on 30th place over 10 km classic, 27th place in the skiathlon and sixth place with the relay and at the Nordic Junior World Ski Championships 2015 in Almaty on 31st place. Place over 10 km freestyle, 25th place in skiathlon and ninth place with the relay. In 2015 he was Swiss junior champion over 15 km, in the pursuit and over 30 km. At the Nordic Junior World Ski Championships in 2016 in Râșnov , he achieved 19th place over 15 km freestyle, seventh place with the relay and fourth place over 10 km classic. In December 2017 he started in Valdidentro for the first time in the Alpencup and finished 29th over 15 km classic and fifth over 15 km freestyle. At the 2017 U23 World Championships in Soldier Hollow , he ran to 37th place in sprint, 21st place in skiathlon and 11th place over 15 km freestyle. At the beginning of the 2017/18 season he made his debut in the World Cup in Davos , where he finished 65th in the 15 km freestyle. In the further course of the season he was in the top ten nine times in the Alpine Cup. He won his first Alpine Cup victory in the mass start race over 15 km freestyle in St. Ulrich am Pillersee and came second in the overall standings at the end of the season. At the 2018 Swiss Championships in Steg , he was third in the sprint and over 15 km and first in the pursuit. At the end of January 2018, at the U23 World Championships in Goms, he achieved 23rd place over 15 km classic, eighth place in the skiathlon and fourth place in the sprint. In December 2018 he got his first World Cup points in Davos with the 27th place in the 15 km freestyle. Klee ran in eleventh place in the 4th stage of the Tour de Ski 2018/19 in Oberstdorf. After finishing 28th in the Ruka Triple at the beginning of the 2019/20 season, he achieved 28th place in the Tour de Ski 2019/20 and 32nd place in the ski tour. At the beginning of March 2020 in Lahti , he achieved second place in the relay, his first podium in the World Cup.