"Mediocrity cannot be the demand of Hamburg," said Fink in an interview with Kicker magazine. "It is also not mine."
Hamburg have not played in the Champions League since 2006 but the coach is keen for his team to get back in Europe's elite club competition.
"We must again eye qualification for international football," said Fink. "In two or three years we want to target the Champions League."
The former Basel coach, who took over on October 13 after Michael Oenning was sacked, has lifted the 1983 European champions off the bottom of the league.
Hamburg are now in 13th place with former Bayern Munich player Fink yet to taste defeat in his eight league games at the helm.
His only defeat came on December 21 when Hamburg were beaten 2-1 by VfB Stuttgart in the German Cup.
Hamburg are the only team never to have played in the second division since the introduction of the Bundesliga in 1963.
Fink's team play their first league game since the winter break when they host Borussia Dortmund on Sunday.
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