The pulsar velocity distribution provides information about the binary history of pulsar progenitors, as well as the asymmetries of the supernova events in which pulsars are born. Studies of local pulsars present a biased view of this distribution, because they preferentially select low-velocity pulsars that have remained near their birthplaces in the Galactic plane. Using the VLA, we have studied the proper motions of a large sample of distant pulsars. These pulsars are generally faint, and the expected proper motions are small. In this paper, we describe the data analysis techniques that we have developed to allow precise astrometric measurements of faint sources with the VLA. These techniques include "gating" the VLA correlator to increase the signal-to-noise ratio of the pulsar by gathering data only during the pulse. Wide-field imaging techniques, including multiband imaging to account for bandwidth smearing, were used to allow identification of multiple in-beam background sources for astrometric calibration. We present the analysis of three pulsars and demonstrate that astrometric accuracy of about 10 mas can be obtained for individual sources with our technique, allowing measurement of proper motions with errors of only a few milliarcseconds per year over our 7 year baseline.