Klien began competitive motorsport in his early teens, racing karts in Austria and Switzerland from 1996 to 1998 and winning the Swiss karting championship in his first year. He moved to Formula BMW ADAC in 1999, finishing fourth overall, and continued in the series through 2001, winning five races and ending third. He transitioned to Formula Renault in 2002, winning the German Formula Renault title with four victories while finishing fifth in the European series.
In 2003, Klien competed in the Formula 3 Euro Series with Mücke Motorsport, winning four races and finishing runner-up to Ryan Briscoe in the championship standings. He also won the 2003 Marlboro Masters of Formula 3 at Zandvoort, one of the season's most prestigious non-championship events. This performance brought him to the attention of Formula One teams and he was signed by Jaguar Racing for the 2004 season before the year was out.
Klien joined Jaguar alongside the more experienced Mark Webber for 2004. Despite the pressure of partnering an established driver, Klien proved reliable — retiring on only four occasions from 18 races — and became the first driver to outqualify Webber from the same team. His sole points-scoring finish of the season came at the Belgian Grand Prix, where he placed sixth for three championship points.
When Red Bull purchased Jaguar at the end of 2004 and renamed it Red Bull Racing, Klien was retained for 2005 alongside David Coulthard. He shared his race seat with Vitantonio Liuzzi during the middle of the season when Red Bull exercised an unusual rotation arrangement, before returning to race consistently. He scored points in the opening two races of the season and took a fifth-place finish in the season finale in China. For 2006, Klien again partnered Coulthard. He started the year well with a top-ten in Bahrain, but was unable to match Coulthard's consistency across the season. Red Bull announced in August 2006 that Mark Webber would replace him for 2007, and Klien was released from the team with three races remaining.
Klien joined Honda as their test and reserve driver for 2007, standing in for the injured Jenson Button during second practice at the British Grand Prix. He signed with BMW Sauber as reserve and test driver for 2008 and 2009. During this period, he also pursued endurance racing, driving for Peugeot at the 2008 24 Hours of Le Mans with Ricardo Zonta and Franck Montagny; the three finished third overall. He returned to Le Mans with Peugeot in 2009, finishing sixth in a car shared with Pedro Lamy and Nicolas Minassian.
Klien joined Hispania Racing (HRT) in 2010, making practice appearances at the Spanish and European Grands Prix before replacing the ill Sakon Yamamoto as a race driver from the Singapore Grand Prix onwards. His Singapore qualifying effort — 22nd, more than a second ahead of teammate Bruno Senna — was a creditable performance given HRT's limited resources. He replaced Yamamoto for the season's final two races in Brazil and Abu Dhabi but was not retained for 2011.
After Formula One, Klien competed in endurance racing, V8 Supercars (including the Bathurst 1000 alongside Russell Ingall), and the European Le Mans Series. He joined the Austrian Armed Forces (Bundesheer) in 2017 while continuing to race in GT machinery, competing in the Deutsche Tourenwagen Masters part-time in 2021 with JP Motorsport's McLaren and in GT World Challenge Europe from 2022. Klien became an expert analyst for ServusTV's Formula One coverage from the 2021 season. He has also written a monthly column for the Pitpass.com website since late 2012.
Klien arrived in Formula One as one of Europe's most promising young drivers after his Formula 3 Euro Series runner-up result and Formula Renault title. His Formula One career, while not yielding the results some predicted, demonstrated consistent professionalism. His best season came in 2005, when a fifth-place finish in China hinted at the potential that his Red Bull exit prematurely cut short. His endurance racing performances with Peugeot — particularly the 2008 Le Mans podium — represent a secondary strand of genuine achievement.