Müller began karting at age twelve in 2004 and spent five years in kart competition before switching to single-seaters in 2010. He joined Eifelland Racing in the ADAC Formel Masters, finishing ninth in the championship in his debut car-racing season. Returning in 2011 with ma-con Motorsport, he took four victories and reached the podium twelve times across the season, ultimately placing third in the standings behind Pascal Wehrlein and Emil Bernstorff.
The 2012 season saw Müller graduate to the FIA Formula 3 European Championship with Prema Powerteam, where he collected three podiums and finished eighth overall. He competed in F3 again in 2013, splitting the campaign between Ma-con and Van Amersfoort Racing and ending ninth with a podium and four further top-five finishes. His training as a mechanic — he worked regularly in his brother's car workshop — gave him an unusually deep technical grounding for a young driver at that level.
Müller's strong formula performances attracted Porsche's attention and he was selected for the Porsche Junioren factory programme for the 2014 season. Racing in the Porsche Carrera Cup Germany with Team Deutsche Post by Project 1, he finished fourth in the championship in his first year and placed seventh overall in the Porsche Mobil 1 Supercup with Team Project 1, taking one Supercup win and earning runner-up in the Supercup rookie standings.
In 2015 he stepped up his Supercup challenge significantly, winning four races — including his first win at the Hungaroring on the Hungarian Grand Prix support card — and finishing second in the overall standings, just behind the champion. He also made a single appearance in the FIA World Endurance Championship that year with Porsche Team Manthey in the LMGTE Pro class at Spa.
His breakout season was 2016. Racing in the Porsche Carrera Cup Germany with Konrad Motorsport, he won ten of sixteen races and took the championship by a commanding margin. Simultaneously, driving the Lechner MSG Racing Team's Supercup entry, he won three rounds and claimed the Porsche Mobil 1 Supercup title. The double German and international Porsche championship in a single season — combined with a class victory at the Dubai 24-hour race — made the case for his promotion and he was elevated to Porsche works driver status for 2017.
Müller gave his works debut in 2017 and quickly began contesting a wide range of GT championships under the Porsche factory umbrella. He raced in the ADAC GT Masters with Precote Herberth Motorsport, the Intercontinental GT Challenge, and the Japanese Super GT GT300 class with D'station Racing, finishing eleventh in the GT300 standings. He also began a recurring relationship with Falken Motorsports at the Nürburgring 24 Hours, a connection that would endure for several years.
His first 24 Hours of Le Mans start came in 2018 with the Porsche GT Team, racing a 911 RSR in the GTE Pro class. The same year he continued in Super GT with D'station Racing, improving to ninth in the GT300 standings with one podium. In 2019 he returned to Le Mans with the Porsche GT Team, finishing seventh in class in the GTE Pro category, and contested the Blancpain GT Series Endurance Cup with Rowe Racing, starting from pole position for one round.
From 2020 onwards Müller focused increasingly on the GT World Challenge Europe and its endurance component. In 2020 he joined Dinamic Motorsport for the GT World Challenge Europe Endurance Cup, winning one round and reaching two further podiums to take third place in the championship — his best result in the series to that point. That year he also took third overall in the Spa 24 Hours.
Through 2021–2023 he continued competing widely, including multiple ADAC GT Masters seasons with Allied-Racing and Team Joos by RACEmotion, regular Falken Motorsports outings at the Nürburgring 24 Hours, and Le Mans Cup entries. In 2024 he moved to Rutronik Racing for the GT World Challenge Europe, winning once in the Sprint Cup and finishing fifth in that championship's standings.
The 2025 season produced his most complete campaign. Racing for Rutronik Racing in the GT World Challenge Europe Endurance Cup, Müller accumulated three podiums — including a second place at Paul Ricard, second at the Spa 24 Hours, and third at the Nürburgring — to take the Endurance Cup drivers' championship. He also finished fifth in the Sprint Cup that year, taking one victory at Zandvoort and further podiums at Magny-Cours, Brands Hatch, and Misano.
The Nürburgring Nordschleife has been a recurring venue throughout Müller's career. He has started the Nürburgring 24 Hours multiple times with Falken Motorsports, achieving a best result of fourth overall in the SP9 class in 2021. In the Nürburgring Langstrecken-Serie he recorded victories in 2022, 2024, and 2025, including the ADAC Westfalenfahrt and ADAC Ruhr Pokal in 2025.
Across more than 450 career race starts, Müller has accumulated 39 victories and over 116 podiums. He holds FIA Gold driver categorisation, reflecting his tenure as a top-tier factory representative. As of 2026 he continues to campaign internationally, racing in Super GT GT300 with Seven x Seven Racing in Japan, the IMSA SportsCar Championship, and the GT World Challenge Europe Endurance Cup with Rutronik Racing, with further Nürburgring Langstrecken-Serie outings planned with Falken Motorsports.