The increasing power densities of high-performance electronic systems require efficient heat dissipation to maintain optimal performance and reliability. Thermal interface materials (TIMs) are widely used to minimize contact resistance between two solid materials in an electronic package for effective heat dissipation. Localized thermal hotspots and the resulting thermally-induced warpage can adversely affect the TIM structure and, thus, TIMs must be able to accommodate mechanical stresses. Hence, both thermal conductivity and mechanical compliance are critical physical properties in the design of TIM. Structural optimization techniques such as topology optimization can generate complex non-conventional geometries, which can achieve the required thermo-mechanical performance. Here, we design a metal-polymer composite TIM utilizing the multi-physics, density-based topology optimization framework. A material interpolation scheme [solid isotropic material interpolation with penalization (SIMP)] is used to find the optimal distribution of the metal and polymer within the given design domain. This enables the design of a composite TIM with an elastic modulus less than 2 GPa and an effective thermal conductivity as high as 50 W/(m~K). Furthermore, two self-competing thermomechanical objective functions [(a) mechanical compliance and (b) temperature variance] are introduced to maximize thermal conductivity for better heat dissipation while ensuring high mechanical compliance. The designed high thermal conductivity TIM structure efficiently dissipates heat from localized thermal hotspot regions in both the transverse and lateral directions while being mechanically compliant to withstand the thermal warpage.