[2] Around the age of five, he began playing ice hockey with his brother, practicing in a backyard rink and on Garner Lake near his family's cabin.

[3][4] A childhood fan of the Edmonton Oilers of the National Hockey League (NHL), he would frequently attend games at Rexall Place with his father.

[6] Selected second overall by the Saskatoon Blades of the Western Hockey League (WHL) in the 2016 bantam draft,[7] Dach immediately signed with the team thereafter.

[8] Following a series of mid-season injuries to the team's forward core, the Blades received permission to promote Dach, an underage player, for the remainder of the 2016–17 season.

[25][20] After sustaining an injury to his wrist during the 2021 World Junior Ice Hockey Championships, Dach subsequently missed much of the pandemic-shortened 2020–21 NHL season.

[27][28] Many observers considered it a poor sign that he was unable to generate offensively despite pairing with high caliber linemates such as Alex DeBrincat and Patrick Kane, while also noting that the Blackhawks' organizational depth had likely "pushed Dach into a role he was not even close to ready for.

[27] During the course of the 2022 NHL entry draft, Dach was traded by the Blackhawks to the Montreal Canadiens in exchange for the ensuing 13th overall pick as well as a third-round selection.

[30] Initially moving defenceman Alexander Romanov to the New York Islanders in order to acquire the higher pick to make said trade,[31][32] Canadiens general manager Kent Hughes stated that he believed Dach could live up to his potential despite his struggles in preceding seasons.

"[40] In his first game back at the United Center in Chicago following the aforementioned trade, he scored the decisive shootout goal, giving the Canadiens a 3–2 win over his former team.

[43] Entering the 2023–24 season with high expectations, Dach was given two new wingers in Juraj Slafkovský, the Canadiens' first overall selection in 2022, and fellow 2019 draftee Alex Newhook, newly acquired from the Colorado Avalanche via trade.

[53] In the gold medal game versus the Swedish national under-18 team, he scored a first period goal, helping lead Canada to a 6–2 victory.

"[61] On January 3, 2025, the brothers competed against each other for the first time in their respective professional careers in a 4–2 Blackhawks win over the Montreal Canadiens at the United Center.