Robertson was born in Arcadia, California, but his family moved to Northville, Michigan, at the age of 10 to increase his opportunities for ice hockey.

Robertson was traded to the Niagara IceDogs partway through the 2018–19 OHL season, during which he recorded 117 points in 62 games to win both the Eddie Powers Memorial Trophy and the CHL Top Scorer Award.

Robertson made Dallas's opening night roster the following season, where he was the Calder Memorial Trophy runner-up with 17 goals and 28 assists in 51 games.

Robertson continued his offensive production during the 2021–22 season, becoming the first player in Dallas Stars history to record hat-tricks in consecutive games.

[2] Beginning with Robertson's older brother Michael, all of Mercedes and Hugh's children were childhood hockey fans, and the family owned season tickets to see the Los Angeles Kings at the Staples Center.

[3] Robertson began playing hockey around the age of four or five,[2] and when he was 10 years old, the family moved to Northville, Michigan, where there were more opportunities for him to ice skate.

[10] He impressed head coach Paul McFarland during rookie camp and made the Frontenacs' 2015–16 opening day roster at 16, the youngest possible age for junior ice hockey.

[11] He recorded his first OHL goal on October 7 against the Oshawa Generals,[12] and on December 11, he scored his first junior hat-trick, leading Kingston to a 4–1 victory over the Sault Ste.

[16] Following his breakout rookie performance, Robertson began the 2016–17 OHL season playing on Kingston's top offensive line with Warren Foegele and Ted Nichol.

[19] He added another five goals and 18 points in 11 postseason games before the Frontenacs were swept by the Peterborough Petes in the second round of OHL playoffs.

[34] Returning to the Frontenacs as an assistant captain,[35] Robertson began the 2018–19 season on an offensive tear: by mid-November, he led the OHL with 22 goals in as many games and was seventh in the league with 33 points.

[37][38] On November 20, the week after receiving these accolades, Robertson and defenseman Jacob Paquette were traded to the Niagara IceDogs for Billy Constantinou, Ian Martin, and 11 draft picks.

[42] In the final week of the regular season, Robertson recorded 11 points in three games to clinch the OHL Central Division title for the IceDogs.

[49] He debuted that night, recording his first NHL assist on Tyler Seguin's goal during the Stars' 3–2 victory over the Toronto Maple Leafs.

[51] By the time that the COVID-19 pandemic suddenly halted both the NHL and AHL regular season, Robertson had 25 goals and 47 points in 60 games for the Texas Stars.

Robertson was named to the team's opening night roster as a fourth-line winger alongside Nick Caamano and Jason Dickinson.

[59] While the Stars narrowly missed the 2021 Stanley Cup playoffs,[60] Robertson finished the season with 17 goals and a rookie-leading 28 assists in 51 games.

He scored the most goals of any player since Jamie Benn during the 2015–16 season, and was only the fourth member of the Stars to reach the 40-goal mark following their relocation to Dallas.

[69][70] The Stars faced the Calgary Flames in the first round of the 2022 Stanley Cup playoffs, with Robertson making his postseason debut in Game 1.

[80] Robertson, joined by Jake Oettinger and Ryan Shea, was one of three young members of the Stars to represent the United States at the 2021 IIHF World Championship in Latvia.

[83] Robertson's younger brother Nick was selected by the Toronto Maple Leafs in the second round of the 2019 NHL Entry Draft.