We consider Feller processes whose generators have the test functions as an operator core. In this case, the generator is a pseudo differential operator with negative definite symbol q(x, ξ). If q(x, ξ) < c(1 + ξ ²), the corresponding Feller process can be approximated by Markov chains whose steps are increments of Lévy processes. This approximation can easily be used for a simulation of the sample path of a Feller process. Further, we provide conditions in terms of the symbol for the transition operators of the Markov chains to be Feller. This gives rise to a sequence of Feller processes approximating the given Feller process.