Born in Dortmund, Klos arrived at Borussia Dortmund in the summer of 1990, from neighbouring TSC Eintracht Dortmund. He made his Bundesliga debut on 4 May 1991 at only 19, in a 2–2 home draw against SG Wattenscheid 09.

On 24 December 1998, Klos signed with Scottish club Rangers on a contract that made him one of the highest-earning players in Europe. He replaced Lionel Charbonnier as first-choice and was nicknamed ‘Der Goalie’, a play on Andy Goram’s ‘The Goalie’ moniker. He won his first Scottish Premier League title that season.

In the 2002–03 season, Rangers won the treble, with Klos again the undisputed starter. He was appointed team captain in July 2004, but six months later picked up a knee ligament injury in training, missing the remainder of the campaign and being replaced by Ronald Waterreus. The Dutchman retained his place even after Klos regained full fitness.

Klos looked set to challenge new signing Lionel Letizi after Paul Le Guen’s arrival at the start of 2006–07, but he suffered a biking injury which gave long-term third choice Allan McGregor the chance to play. On 22 February 2007, he made his first appearance of the season for the first team against Hapoel Tel Aviv FC in the round of 32 of the UEFA Cup, after McGregor was sent off (4–0 home win).

In 2009, Klos was inducted into the club’s Hall of Fame.