The initial and boundary conditions of the Galactic star formation in molecular clouds are not well understood. In an effort to shed new light on this long-standing problem, we measured the properties of dense cores and filamentary structures in the Vela C molecular cloud, observed with Herschel. We used the hires algorithm to create high-resolution surface densities (11.7″) from the Herschel multiwavelength dust continuum. We applied the getsf extraction method to separate the components of sources and filaments from each other and their backgrounds before detecting, measuring, and cataloging the structures. The cores and filamentary structures constitute 40% of the total mass of Vela C; most of the material is in the low-density molecular background cloud. We selected 570 reliable cores, of which 149 are the protostellar cores and 421 are the starless cores. Almost 78% (329 of 421) of the starless cores were identified with the gravitationally bound prestellar cores. The exponent of the CMF (α = 1.35) is identical to that of the Salpeter IMF. The high-resolution surface density image helped us determine and subtract backgrounds and measure the sizes of the structures. We selected 68 filaments with at least one side that appeared not blended with adjacent structures. The filament widths are in the range from 0.15 pc to 0.63 pc, and have a median value of W = 0.3 ± 0.11 pc. The surface densities of filaments are well correlated with their contrasts and linear densities. Within uncertainties of the filament instability criterion, many filaments (39) may be both supercritical and subcritical. A large fraction of filaments (29), in which are found 94 prestellar cores, 83 protostellar cores, and only 1 unbound starless core, can definitely be considered supercritical. Taking into account the uncertainties, the supercritical filaments contain only prestellar and protostellar cores. Our findings support the idea that there is a direct relationship between the CMF and IMF and that filaments play a key role in the formation of prestellar cores, which is consistent with the previous Herschel results.