Born in Horsens, Kjær started playing football with small-time club Lund IF near Horsens, before moving to the youth team of Danish Superliga club FC Midtjylland in 2004. He was named the best player of a French youth tournament in April 2006, and Midtjylland subsequently rejected offers from Lille OSC for Kjær and Jesper Juelsgård. He made his senior debut for Midtjylland in the Viasat Cup training matches in June 2006, and was promoted to the first team squad in January 2007. He had a successful trial with Real Madrid in August 2007, but their transfer offer was refused by FC Midtjylland. He signed a five-year contract with Midtjylland instead in September 2007.
He made his Superliga debut on 30 September 2007, in the 2–0 victory against AGF, and played four games in the first half of the Danish Superliga 2007-08 season. By October 2007, Kjær had attracted the likes of Premier League clubs Middlesbrough and Chelsea. In February 2008, Kjær signed a five-year contract with Italian club Palermo, effective from the summer 2008, earning Midtjylland a €4 million transfer fee. He played all 15 games in the second half of the Superliga season, forming a defensive partnership with Magnus Troest.
In July 2008, Kjær joined Italian club Palermo.
He made his Palermo debut on 26 October 2008 as a half-time substitute against Fiorentina in Serie A. His performance won praise despite Palermo's 3–1 home defeat.
He scored his first goal for Palermo in his third appearance on 2 November in a 3–0 win over Chievo. He ended the 2008–09 season as Palermo's top scoring defender in Serie A with three goals.
Kjær confirmed his reputation as a first team regular in the 2009–10 season, and formed a strong defensive partnership with fellow centre-back Cesare Bovo. In September 2009, he signed an improved contract with the Sicilian club. He was named 2009 Danish Talent of the Year in November 2009, beating Arsenal's Nicklas Bendtner and Ajax's Christian Eriksen after his successful first season in Palermo. In January 2010, he was also named as Danish Footballer of the Year for 2009. The next day, he celebrated with a goal against Cagliari that decided the match for Palermo as the match ended 2–1. He scored twice during the season, and played a total 35 Serie A games.
Clubs like English side Manchester United, Manchester City, Tottenham Hotspur, Italian side Juventus had reportedly shown interest in signing the player.
On 30 June 2010, Wolfsburg confirmed they had opened talks with Kjær. Wolfsburg had beaten off competition from various European heavyweights to sign Kjær. On 8 July 2010, Wolfsburg signed Kjær for an undisclosed fee believed to be in excess of £10million. Following his move to Wolfsburg, Kjær said: "The transfer means I have the chance to play for a top club, I think this is the right step for me. We have a strong team that can challenge at the top of the Bundesliga. First, I need to secure a regular place." The deal made Kjær the most expensive Danish footballer in the world. Making the shift from Serie A football to the Bundesliga meant that Kjær would become one of the best paid Danish footballers with an annual salary of around €3 million. The move from Serie A to the Bundesliga and VfL Wolfsburg was not only economically justifiable for the young Danish defender. Both national coach Morten Olsen and Kjær himself said after the shift that this is both sporty and developmentally was a step in the right direction for him.
Kjær made his unofficial debut for Wolfsburg in a pre-season fixture against FC Augsburg, in which Wolfsburg ran out 1–0 winners.
Kjær made his Bundesliga debut for Wolfsburg in a 2–1 loss against Bayern Munich on 20 August 2010.
After a poor start to the season for Wolfsburg, with varying results and unmet expectations; Kjær scored his first goal for Wolfsburg in a 2–0 win over Stuttgart when he headed the ball in over the line after a corner kick. Several statisticians have noted that the ball took a deflection, however the goal was eventually credited to Edin Džeko. Despite much criticism from the media for his poor performances, Kjær played in the starting lineup in all 17 Bundesliga matches in the autumn 2010 season. After having formed a successful partnership with Italy international Andrea Barzagli in Wolfsburg's central defence, Kjær would then feature alongside German international Arne Friedrich. In his 19th appearance in the league, Kjær scored his first Bundesliga goal for the club. Kjær was match winner in the 0–1 away game against Mainz 05, with a headed goal in the 83rd minute, which Kjær secured the first Wolfsburg victory in eight matches – after seven drawn games in a row.
Following Steve McClaren's sacking (who brought him to the club) due to further poor results in February 2011, Felix Magath returned to Wolfsburg (the manager guided the club to win their first Bundesliga title). Magath made a comment on criticising the 'poor condition' of the players he inherited (including Kjær) from McClaren. During the English coach had Kjær started 20 out of 21 Bundesliga matches, after having been out with an injury in round 20 against Dortmund. In connection with the dismissal declared affection in public, that he was sorry for firing, and that he had been happy to work with Steve McClaren. In his first season, Kjær made 32 appearances and scored two goals.
In his second season at Wolfsburg, Kjær made three appearances for Wolfsburg. Kjær's time at Wolfsburg has been met by frustration, as the club narrowly avoided the drop last season and he has come under fire for some lacklustre displays.
In the summer transfer window 2011, Kjær revealed that head coach Magath told him that he'll be allowed to leave this summer and reported that he had fallen out favor with Magath. Italian side Roma revealed the club are on the verge of signing Kjær on a €10 million move from Wolfsburg as a replacement for centre-back Philippe Mexès who left for AC Milan on a free transfer. However, the move to Roma was put back on hold after reports in Italy claimed the defender had been recalled by his club Wolfsburg. Instead on 30 August 2011, Kjær went out on a season-long loan deal to Roma, with the option to purchase at the end of the season.
Kjær made his debut for Roma in a 0–0 draw against Inter Milan on 17 September 2011. On 16 October 2011, Kjær received a red in a 2–1 loss against Lazio. Roma director Walter Sabatini dropped a strong hint that the club may decide against signing Kjær on a permanent basis due to his poor performance at Roma. He has been largely disappointing for Roma after he was to blame after Roma's 4–2 defeat to Cagliari on 1 February 2012. In the end, Roma decided against signing Kjær permanently, stating that he had made too many costly errors.
After his loan-spell at Roma ended, Kjær stated he will not return to Wolfsburg, after coach Magath decided the Dane was not going to be part of his first team plans for the club's 2012–13 season. However, Magath soon after announced he was now willing to bury the hatchet with Kjær and would be keen to bring him into his first team at Wolfsburg.
On 5 July 2013 it was announced that Kjær had signed a 4-year deal with Ligue 1 side, Lille. Kjær impressed during his first season in the French Ligue 1 as leader of the best defense in the major European national leagues in the 2013–14 season conceding only 26 goals in 38 matches including an impressive number of 21 clean sheets. As a result of the solid performances Kjær frequently appeared on various medias team of the week during the entire season, eventually earning a spot on "the Ligue 1 Team of the Season" alongside teammate and goalkeeper Vincent Enyeama. He scored his first ever Ligue 1 goal against Lorient with a deflected free kick.
On 16 June 2015, Kjær joined Turkish side Fenerbahçe signed four-year deal for €7.65 million.
Kjær made his debut for the Denmark national under-19 football team in September 2006, at the age of 17. He played a total 10 games and scored one goal for the under-19s until October 2007, and was named 2007 Danish under-19 talent of the year as the under-19s coach found Kjær had shown himself a talented defender despite his young age. Following his breakthrough with Midtjylland, Kjær was rested from the under-19s in March 2008 to avoid wearing him out.
He was called up for the Denmark national under-21 football team in May 2008 as a replacement for Michael Jakobsen, making his debut in a 0–4 defeat against the Germany under-21s. He was out of the under-21 team during his first time at Palermo, but was recalled in October 2008, spending the game as an unused substitute to Magnus Troest and Mathias Jørgensen. Kjær was called up to the senior Danish national team in February 2009, as national team coach Morten Olsen was impressed by his form for Palermo, and saw him and Per Krøldrup as main competitors to replace the retired defender Martin Laursen. He was an unused substitute for the friendly match against Greece, and returned to the under-21s for a single game in March 2009, playing well enough that under-21 coach Keld Bordinggaard recommended him for a recall to Olsen's squad.
He was once again a part of the senior squad for the 2010 FIFA World Cup qualification game against Sweden on 6 June 2009. Per Krøldrup was unavailable due to injury, and Kjær played the full game despite an injury of his own, as Denmark won 1–0. Kjær established himself in the Danish starting line-up, played the following three qualification games, and helped Denmark qualify for the 2010 World Cup. For the September 2009 qualification game against Portugal, Kjær told reporters that Cristiano Ronaldo should be stopped by a series of small fouls and one hard tackle. This prompted a complaint to FIFA by the Portuguese Football Federation requesting Kjær be suspended for unsportsman-like conduct, but the case was dropped following the mediation of the Danish Football Association.
On 28 May 2010, Denmark coach Morten Olsen announced that the player would be part of the final squad of 23 participating in the 2010 FIFA World Cup in South Africa. He was in the starting eleven in Denmark's first two games against the Netherlands and Cameroon, where he received a yellow card in both games and therefore missed the third and crucial group game against Japan. However, in the match against Cameroon he was highly praised for his excellent long-range passing which helped to set up Nicklas Bendtner's first goal.
On 7 September 2010 following Denmark win over Iceland, Stig Tøfting has not been impressed by Kjær and criticized him that he is making too many mistakes. This led Kjær offended and hit back on Tøfting saying "These statements hit me personally! It is the first time that I have come under pressure and I try to take these words as a motivation. I am not happy about his (Tøfting's) opinion. He just says that to stay in the public attention. He should be serious!"
Kjær was a part of the Danish UEFA Euro 2012-team, and was among the starting 11 in each of the three group games against the Netherlands, Portugal and Germany.Statistics accurate as of last match played on 29 October 2016.
Scores and results list Denmark's goal tally first.
Danish U19 Player of the Year: 2007
Danish Talent of the Year: 2009
Danish Player Of The Year: 2009