Klien developed his enthusiasm for motorsport after his father took him to watch a go-kart race, and he credits a meeting with Ayrton Senna as a further inspiration. He began karting at 13, winning the Swiss karting championship in his first year of competition (1996), and transitioned to Formula BMW in 1999. He progressed to German Formula Renault, where he won the title in 2002, then moved to the Formula Three Euroseries with Mücke Motorsport, winning four races and finishing runner-up to Ryan Briscoe. He also won the prestigious 2003 Marlboro Masters event at Zandvoort.
Klien was signed by Jaguar Racing for the 2004 season alongside Mark Webber. Despite being under significant pressure as the junior partner to the more experienced Webber, Klien distinguished himself as the first driver ever to outqualify Webber from within the same team. His only points-scoring finish in 2004 came at the Belgian Grand Prix, where he finished sixth for three championship points.
At the end of 2004 Red Bull acquired Jaguar and renamed the team Red Bull Racing.
In 2005, Klien and Vitantonio Liuzzi shared the second race seat at Red Bull alongside David Coulthard in an unusual mid-season arrangement. Klien held the seat for the first part of the season, yielded it to Liuzzi for four rounds, then reclaimed it for the remainder. His best results that year were a fifth-place finish at the season-ending Chinese Grand Prix and an eighth in Turkey.
After a confirmed full-time seat for 2006 alongside Coulthard, Klien began the year strongly with a top-ten qualifying effort at the first two rounds. He scored one point in Bahrain before a first-lap collision with Kimi Räikkönen in Malaysia compromised his season. He was outperformed overall by Coulthard, who gave the team its first podium that year. In August 2006, Red Bull confirmed Klien would not continue as a race driver, with Mark Webber signed in his place. Klien was released for the final three races of the season and replaced by test driver Robert Doornbos.
Klien joined Honda as test and reserve driver for 2007. He stood in for the injured Jenson Button during a free practice session at the British Grand Prix. Honda later released him to test for Spyker midseason, but the race seat there went to Sakon Yamamoto.
In 2008 and 2009, Klien served as BMW Sauber's test and reserve driver. He also raced in endurance events during this period, finishing third overall at the 2008 24 Hours of Le Mans in a Peugeot 908 HDi FAP shared with Ricardo Zonta and Franck Montagny.
Klien returned to a Formula One race weekend in 2010 with HRT, substituting for the ill Sakon Yamamoto at the Singapore Grand Prix — his first race start since the 2006 Italian Grand Prix. He qualified 22nd and was running ahead of teammate Bruno Senna before a mechanical problem ended his race. He also replaced Yamamoto for the final two races of the season in Brazil and Abu Dhabi.
After leaving Formula One, Klien built a career across multiple disciplines: endurance racing in the European Le Mans Series, V8 Supercars outings including the Bathurst 1000 with Russell Ingall, and GT3 racing for JP Motorsport in the Deutsche Tourenwagen Masters (2021) and GT World Challenge Europe (2022). He joined the Austrian Armed Forces (Bundesheer) in 2017 and has combined military service with racing. Since 2021 he has worked as an expert analyst for ServusTV's Formula One broadcasts.