Efficient and robust numerical methods for two-phase boiling flows are essential in various engineering applications. A common challenge in these methods arises from the presence of jump conditions across the fluid interface due to phase-change. To accurately capture these jump conditions, numerical methods often rely on sharp-interface approaches, such as the ghost fluid method (GFM). An alternate, simpler approach, commonly known as the delta method, utilizes the whole domain formulation (WDF), which models these jumps as singular source terms with Dirac distributions placed at the fluid interfaces . However, it has been demonstrated that the delta method, due to the smoothing of jump conditions, is less accurate than the sharp-interface methods. In the present work, a hyperbolic solver is developed for the WDF with the singular source terms included in the non-conservative system. Hyperbolicity of the system is ensured using a weakly compressible model leading to a fully-explicit solver for boiling flow simulations. A novel HLLC-type path-conservative scheme, which models the fluid interface as a contact wave, is developed to solve the non-conservative system. The HLLC Riemann solver accurately captures the phase-change induced Rankine-Hugoniot jumps across the interface, which were verified using the generalized Riemann invariant analysis. The phase-field method is adopted to capture the interface dynamics, as the diffused nature of the interface aids in accurate estimation of interface normals and curvatures using simple gradient estimation techniques. The path-conservative method is first tested on problems with fixed phase-change mass flow rate to assess the solver's capability to accurately predict the interface movement and the jumps across it. The solver is further extended to include the energy (temperature) equation to solve benchmark boiling flow problems. Simulations on both structured and unstructured grids demonstrate the robustness and adaptability of the proposed method.