Covellite (CuS) is an important mineral sulfide that can be used in many technological applications. It has a simple formula but a complex structure consisting of alternating layers of planar CuS3 triangles and CuS4 tetrahedrons with S-S bonds. Accurate first-principles calculations are performed for covellite structure (CuS), aiming to provide insights about its structural, mechanical and electronic properties and to unveil the nature of its chemical bonding. DFT and DFT+U methods have been used and showed to be sensitive to the correlation treatment (U value). Although it is not possible to extract a universal value of the U, this study indicates that U = 5 eV is an adequate value. The electronic structure analysis shows a significant metallic character due to p(S)-d(Cu) orbital interactions up to Fermi level. The projected density of states indicates that most of the contribution comes from the atomic orbitals in the [001] plane of the covellite, explaining the conductivity anisotropy observed experimentally. Topological analysis of the electron density was performed by means of quantum theory of atoms in molecules (QTAIM). Two different topological charges in Cu and S were calculated, confirming an ionic model with mix-charges. This mineral presents ionic degree of similar to 32%. On the basis of the QTAIM analysis, the covalent character of S S bond is confirmed, and the favored cleavage of CuS at the [001] surface might be at the Cu-S bond. The S atoms occupy most of the cell volume, and their contributions dominate the crystal compressibility: k(S) approximate to k(CuS).