In recent years, there has been a growing interest in the application of the artificial compressibility approach to solve incompressible Navier-Stokes equations. This method proves particularly valuable when simulating multiphase flows, as it enables the simultaneous solution of the Navier-Stokes equations and the interface advection equation in a tightly coupled manner. A noteworthy advantage of the artificial compressibility method, distinguishing it from other approaches, lies in its ability to facilitate the utilization of advanced techniques based on the solution of Riemann problems. In the context of fluid-fluid interfaces resembling contact waves, the artificial compressibility approach provides an exact representation of these phenomena through the application of Riemann solvers. One of the most popular Riemann solvers is the Harten-Lax-van Leer with contact (HLLC) solver. In the present work, we developed a convective flux formulation based on the HLLC approach for the coupled system of incompressible Navier-Stokes equations and the level set advection equation in the artificial compressibility framework. The numerical solution of the governing system of equations is carried out using a cell-centered finite volume approach on an unstructured triangular mesh. The efficacy of the proposed formulation is demonstrated by solving a set of several standard incompressible two-phase flow test problems involving gravitational, viscous and surface tension forces.