Using Hubble Space Telescope/ACS H{alpha} images of SNR 0509-67.5 taken ~10yr apart, we measure the forward shock (FS) proper motions (PMs) at 231 rim locations. The average shock radius and velocity are 3.66+/-0.036pc and 6315+/-310km/s. Hydrodynamic simulations, recast as similarity solutions, provide models for the supernova remnant's expansion into a uniform ambient medium. These are coupled to a Markov chain Monte Carlo (MCMC) analysis to determine explosion parameters, constrained by the FS measurements. For our baseline model, the MCMC posteriors yield an age of 315.5+/-1.8yr, a dynamical explosion center at 5:09:31.16,-67:31:17.1 and ambient medium densities at each azimuth ranging over 3.7-8.0x10^-25^g/cm^3^. The age uncertainty can be an order of magnitude larger when considering other models, or subsets of the data. We detect stellar PMs corresponding to speeds in the Large Magellanic Cloud >=770km/s. Five stars in the remnant show measurable PMs but none are moving radially from the dynamical center. There are four stars 1.4" from the center, including three faint, previously unidentified ones. Using coronal [FeXIV]{lambda}5303 emission as a proxy for the reverse shock location, we constrain the explosion energy (for a compression factor of 4) to a value of E=(1.30+/-0.41)x10^51^erg for the first time from shock kinematics alone. Higher compression factors (7 or more) are strongly disfavored based on multiple criteria, arguing for inefficient particle acceleration in the Balmer shocks of SNR 0509-67.5.