We present measurements of internal proper motions at more than 500 positions of NGC 2392, the Eskimo Nebula, based on images acquired with WFPC2 on board the Hubble Space Telescope at two epochs separated by 7.695 yr. Comparisons of the two observations clearly show the expansion of the nebula. We measured the amplitude and direction of the motion of local structures in the nebula by determining their relative shift during that interval. In order to assess the potential uncertainties in the determination of proper motions in this object, in general, the measurements were performed using two different methods, used previously in the literature. We compare the results from the two methods, and to perform the scientific analysis of the results we choose one, the cross-correlation method, because it is more reliable. We go on to perform a "criss-cross" mapping analysis on the proper motion vectors, which helps in the interpretation of the velocity pattern. By combining our results of the proper motions with radial velocity measurements obtained from high resolution spectroscopic observations, and employing an existing 3D model, we estimate the distance to the nebula to be 1.3 kpc.