A combination of low-energy electron diffraction, x-ray and ultraviolet photoelectron spectroscopy, and scanning-tunneling microscopy studies, in conjunction with ab initio calculations leads us to suggest a model for the carbon (C)-induced $\mathrm{Si}(001)c(4\ifmmode\times\else\texttimes\fi{}4)$ atomic structure. This surface superstructure is obtained in a defined range of ${\mathrm{C}}_{2}{\mathrm{H}}_{4}$ exposures at 600 \ifmmode^\circ\else\textdegree\fi{}C. Experimental probes show that the $c(4\ifmmode\times\else\texttimes\fi{}4)$ superstructure involves C atoms in both surface and subsurface sites. This is reflected in well-marked features in photoemission valence- and core-level spectra. Surface carbon atoms are stabilized in Si-C heterodimers, with a surface density of about 0.25 monolayer (ML) [i.e., two C atoms per $c(4\ifmmode\times\else\texttimes\fi{}4)$ unit cell of eight atoms]. In the subsurface region, carbon atoms substitute for Si atoms in well-defined sites of the third or fourth layers of the Si substrate. The subsurface C density increases with ${\mathrm{C}}_{2}{\mathrm{H}}_{4}$ exposure time up to a limit value of about 0.5 ML, within the $c(4\ifmmode\times\else\texttimes\fi{}4)$ surface structure. Further exposure disrupts the $c(4\ifmmode\times\else\texttimes\fi{}4)$ reconstruction and leads to a $(2\ifmmode\times\else\texttimes\fi{}1)$ low-energy electron diffraction pattern. Interaction with atomic hydrogen shows that the surface contains a mixture of heterodimers (Si-C) and homodimers (Si-Si), with an 1:1 proportion. These assignments are supported by first-principle calculations, which yield valence band and core level states in fairly good agreement with the experiment. Furthermore, total energy calculations strongly favor C incorporation in surface Si-C dimers and in third and fourth layer sites, and rule out C incorporation in sites of the second Si layer. The most stable $c(4\ifmmode\times\else\texttimes\fi{}4)$ surface configuration, suggested by our calculations, consists of alternate Si-C and Si-Si dimer lines. In such a configuration, surface carbon atoms in Si-C dimers induce a surface stress that leads to charge redistribution and atomic relaxation of the adjacent Si-Si dimers, consistent with scanning-tunneling microscopy images. Additional C atoms (in excess of those accommodated in surface sites) are forced in selected compressive (\ensuremath{\alpha}) sites of the third and fourth layers. This model is discussed with respect to the previous models published in the literature.