We report the results of a five-year survey of the northern sky to search for point sources of high energy neutrinos. The search was performed on the data collected with the AMANDA-II neutrino telescope in the years 2000 to 2004, with a live time of 1001 days. The sample of selected events consists of 4282 upward going muon tracks with high reconstruction quality and an energy larger than about 100 GeV. We found no indication of point sources of neutrinos and set 90% confidence level flux upper limits for an all-sky search and also for a catalog of 32 selected sources. For the all-sky search, our average (over declination and right ascension) experimentally observed upper limit Phi(0)=(E/1 TeV)(gamma)center dot d Phi/dE to a point source flux of muon and tau neutrino (detected as muons arising from taus) is Phi(nu mu)+nu(0)(mu)+Phi(nu tau)+nu(0)(tau)=11.1x 10(-11) TeV-1 cm(-2) s(-1), in the energy range between 1.6 TeV and 2.5 PeV for a flavor ratio Phi(nu mu)+nu(0)(mu)/Phi(nu tau)+nu(0)(tau)=1 and assuming a spectral index gamma=2. It should be noticed that this is the first time we set upper limits to the flux of muon and tau neutrinos. In previous papers we provided muon neutrino upper limits only neglecting the sensitivity to a signal from tau neutrinos, which improves the limits by 10% to 16%. The value of the average upper limit presented in this work corresponds to twice the limit on the muon neutrino flux Phi(nu mu)+nu(0)(mu)=5.5x10(-11) TeV-1 cm(-2) s(-1). A stacking analysis for preselected active galactic nuclei and a search based on the angular separation of the events were also performed. We report the most stringent flux upper limits to date, including the results of a detailed assessment of systematic uncertainties.