The metal-semiconductor interface is fundamental to any semiconductor device and the success of advanced technology nodes critically depends upon the minimization of the contact resistance at the interface. In this paper, we calculate the electronic structure of a metal-semiconductor interface (TiGe/Ge contact) within the framework of first-principles density functional theory simulations. We report the modulation of the Schottky barrier height with respect to the different phases of TiGe metal and different crystallographic orientations of Ge substrate. We further compute the I-V characteristics of the TiGe/Ge contact with nonequilibrium Green's function formalism, using a two-terminal device configuration. The calculated transmission spectrum allows us to extract the contact resistance at the metal-semiconductor interface. Furthermore, the onset of Ohmic contact for p-doped TiGe/Ge interface is identified by studying the IV characteristics as a function of increasing active carrier concentration. We find that a doping concentration of 1e21 is sufficient to transform the Schottky contact into Ohmic and thereby achieve a least possible contact resistance at the interfaces. Our paper thus provides useful physical insights into the nanoscale details of the TiGe/Ge interfaces and can guide further process development to minimize the contact resistance.