Raised in a flat in western Amsterdam, Bergkamp was one of four sons of a Catholic plumber. He was never known to socialize much outside of his own family. His parents, who were football fanatics, named him after Manchester United, Manchester City and Scotland striker Denis Law.

Bergkamp was brought up through Ajax's famous youth system, joining the club at age 12. He was given his professional debut by coach Johan Cruijff on December 14, 1986 against Roda JC, and went on to make 14 appearances that season. He played as a substitute in the 1987 European Cup Winners Cup final against Lokomotive Leipzig, which Ajax won. The following season, Bergkamp became a regular for Ajax, winning the Dutch league in 1990, the UEFA Cup in 1992 and the KNVB Cup in 1993. From 1991 to 1993 Bergkamp was top scorer in the Eredivisie, and he was voted Player of the Year in 1992 and 1993. In all, Bergkamp scored 122 goals in 239 games for his hometown club.

In the summer of 1993, Bergkamp was signed by Internazionale of Milan for £12,000,000 alongside his Ajax team-mate Wim Jonk. However, his time in Italy was less successful. Although he won a second UEFA Cup in 1994, he found it hard to adapt to the Italian defensive style of play, scoring just 11 times in 50 appearances. Bergkamp's poor form was not helped by his frosty relationship with the Italian press, and indeed some of his teammates.

After two unhappy seasons at Inter, Bergkamp was signed by Arsenal boss Bruce Rioch in June 1995 for £7.5m. Bergkamp made his debut against Middlesbrough on 19 August 1995 but had to adapt to the English style of play; it took him seven games before he managed to score his first goal, against Southampton. Slowly but surely, Bergkamp's stature grew, playing as a forward behind the main striker, Ian Wright. They soon formed a strong partnership which was one of the finest in the Premier League. Bergkamp has been regarded by many football pundits such as Alan Hansen as being the greatest foreign player to grace the English game.

Bergkamp's arrival at Arsenal was significant, not only as he was one of the first world-class foreign players to join an English club since the lifting of the Heysel ban in 1990, but also because he was a major contributor to the club's return to success after a brief period of mediocrity of the mid-1990s.

Bergkamp hit his best form for Arsenal after the arrival of Arsène Wenger in September 1996. Arsenal won a Premiership and FA Cup double in the 1997-98 season (although Bergkamp missed the cup final with an injury), and Bergkamp was voted PFA Player of the Year. In September 1997, he became the first and so far only player to have come first, second and third in Match of the Day's Goal of the Month competition, for a hat trick against Leicester City. He scored 16 times that season, as well as being involved in setting up many more. At the end of that same season, he helped the Netherlands to a fourth-place finish at the 1998 World Cup. In that competition he scored one of the greatest goals of all time in a quarter final win against Argentina, after receiving a 60 yard pass from Frank de Boer.

In 2005, due to Arsenal's reluctance in offering him a new deal, there was speculation that Bergkamp would leave the club, and possibly football altogether. Bergkamp had said he would retire from football if not offered a new contract with Arsenal for the 2005-06 campaign, despite interest from his former club Ajax. Following Arsenal's penalty shootout victory over Manchester United in the FA Cup Final in Cardiff though, it was revealed he would sign a one-year contract extension, which was custom by the manager Arsène Wenger for players over 30 years of age, keeping him at the London club for their final season at the Arsenal Stadium at Highbury. On April 15, 2006, the home match against West Bromwich Albion, the Arsenal supporters dedicated the original "Supporter's Day" match theme to "Dennis Bergkamp Day", to commemorate Bergkamp's time at Arsenal. Bergkamp came on as a second half substitute, setting up the winning Robert Pirès goal, after Albion had equalised just after he came on. He scored a curler himself at the 89th minute, as Arsenal ran out 3-1 winners. That goal proved to be the last goal Bergkamp scored in Arsenal colours and in competitive football.

Bergkamp officially retired from competitive football after the 2006 Champions League final game against FC Barcelona on May 17, 2006. The last game he played in Arsenal colours was also the last game at Highbury, against Wigan Athletic on May 7, 2006 - the final Premier League game of that season. Bergkamp scored 120 goals for Arsenal in 424 appearances, and set up a further 166 goals in his Arsenal career. This makes him the top assist maker in Premier League history.

Bergkamp made his international debut for the Netherlands against Italy on September 26, 1990. His first major tournament was Euro 92, where the Dutch were defending champions. Bergkamp impressed, but the Dutch lost on penalties in the semi-finals to the team that would go on to win the tournament, Denmark, despite Bergkamp scoring both in regular time and in the penalty shoot-out. Bergkamp also impressed in the 1994 FIFA World Cup, playing in all of his side's matches, scoring a spectacular goal against Brazil, a match the Dutch ultimately lost 3-2. The Netherlands disappointed in Euro 96, with the squad riven by in-fighting, although Bergkamp still scored once, and set up Patrick Kluivert's consolatory goal against England that got the side into the quarter-finals. In the 1998 FIFA World Cup, Bergkamp scored three times, the most memorable of which being the winning goal in the final minute of the quarter-final against Argentina. The Netherlands joint-hosted Euro 2000 and were one of the favourites. After progressing through the "group of death", they lost on penalties to Italy in the semi-finals. Bergkamp didn't score at all during the tournament but still played an important role. After the defeat, Bergkamp announced his retirement from international football. He ended his international career first on the all-time list of goal scorers for the Dutch national team, with 37 goals in 79 games, although his record has since been passed by Patrick Kluivert.

He was selected by Pelé as one of the FIFA 125 greatest living players and is widely regarded as one of the English Premier League's all-time greatest players. In 2007 he was inducted into the English Football Hall of Fame, the first ever Dutch player to receive the honour.