In this paper, a finite element method based on the characteristic for the incompressible Navier-Stokes equations is proposed by introducing Runge-Kutta method. At first, coordinate transformation operation is performed to obtain the alternative Navier-Stokes equations without convection term. Then, instead of the classical characteristic-based split (CBS) method, we use the third-order Runge-Kutta method along the characteristic to carry out time discretization in order to improve calculation accuracy, and segregate the calculation of the pressure from that of the velocity based on the momentum-pressure Poisson equation method. Finally, some classical benchmark problems are used to validate the effectiveness of the present method. Compared with the classical method, the present method has lower dissipation, larger permissible time step, and higher time accuracy. The code can be downloaded at DOI: 10.13140/RG.2.2.36336.56329.