Firstly, a difference scheme with the second-order accuracy is established for the biharmonic equation, thus proving the uniqueness of the solution of this difference scheme, with the convergence and stability further demonstrated by using the extremum principle. The double Richardson extrapolation method is adopted to extrapolate the difference scheme with two sequential extrapolations, resulting in the fourth and the sixth order accuracy at the inner nodes, respectively. The research results indicate that the proposed extrapolation method exerts a significant effect on the improvement of the accuracy of differential scheme solutions under certain conditions.