Bidimensional spectroscopy of the circumnuclear region (central 10" x 8") of the Seyfert 2 galaxy NGC 1068 has been performed by means of an optical fiber system. The 95 spectra obtained simultaneously include the [N II] λλ6548, 6583, Hα, and [S II] λλ6716,6731 emission lines. In the inner regions, the velocity field obtained by cross-correlating the spectra shows a strong S-distortion, which can be explained as due to the presence of radial motions outside the galaxy's plane. The optical nucleus is displaced with respect to the kinematic center. The spectra show multicomponent line profiles due to the integration of several gaseous systems along the line of sight. By using a model of four components, which fits the line profiles obtained remarkably well, we accomplished a deprojection of the corresponding kinematic systems, which allows the velocity fields and line intensity maps of each system to be studied separately. The AGN-like ionizing radiation field seems to exhibit a biconical morphology, judging from the individual line intensity maps associated with the distinct gaseous systems. This is in better agreement with current active galactic nucleus (AGN) models for a Seyfert 2 than the single conical structure proposed by other authors on the basis of filter imaging maps. This result is supported by independent kinematic (i.e., the existence of outflowing cones) and geometric (i.e., small inclination of the cones with respect to the disk) arguments. We propose that the hidden nucleus of NGC 1068 may be located toward northeast of positions recently quoted. On the one hand, the origin of the outflowing gas dominating the inner velocity field, which is likely to be related directly to the active nucleus, is about 1" northeast of the optical nucleus. The center of rotation derived from the outer velocity field also sits northeast of the optical nucleus. On the other hand, the bipolar morphology of the intensity maps of the distinct gaseous system is centered close to the kinematic center, also supporting this conclusion. The proposed location for the hidden nucleus is neither in positional agreement with the radio peak nor with the peak at midinfrared images, according to current astrometry. We have reinterpreted the ionization map [O III]/(Hα + [N II]) and Hubble Space Telescope images of NGC 1068 obtained by other authors, finding that they are compatible with a biconical ionization morphology, and with the proposed location for the hidden nucleus.