In a first step the Chebyshev collocation method for solving the one- dimensional Poisson equation with homogeneous boundary conditions is developed and it is shown that the unknown coefficients of a truncated expansion of the solution with respect to Chebyshev polynomials can be found as the solution of an upper triangular linear system of equations. This result is used to present the Chebyshev pseudospectral approach to the two-dimensional Poisson equation in a square with homogeneous Dirichlet boundary conditions. The coefficients of a truncated double Chebyshev series can be determined from a linear system of equations with a matrix that is given by a tensor product. There result two uncoupled systems of quasi-tridiagonal type for the even and odd coefficients. In case of time dependent problems it is shown how the necessary solution step can be performed by simple matrix products if the inverse formula is used.