A computational framework is presented for the simulation of eukaryotic cell migration and chemotaxis. An empirical pattern formation model, based on a system of nonlinear reaction-diffusion equations, is approximated on an evolving cell boundary using an arbitrary Lagrangian Eulerian surface finite element method (ALE-SFEM). The solution state is used to drive a mechanical model of the protrusive and retractive forces exerted on the cell boundary. Movement of the cell is achieved using a level set method. Results are presented for cell migration with and without chemotaxis. The simulated behavior is compared with experimental results of migrating Dictyostelium discoideum cells.