  Using HST narrow-band H{\alpha} images of supernova remnant 0509-67.5 taken
{\sim}10 years apart, we measure the forward shock (FS) proper motions (PMs) at
231 rim locations. The average shock radius and velocity are 3.66 {\pm} 0.036
pc and 6315 {\pm} 310 km s{^{-1}}. Hydrodynamic simulations, recast as
similarity solutions, provide models for the 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
specific global parameters, the MCMC posterior distributions yield an age of
315.5 {\pm} 1.8 yr, a dynamical explosion center at 5h09m31s.16s
-67{^\circ}31{^\prime}17.1{^{\prime\prime}} and ambient medium densities at
each azimuth ranging from 3.7-8.0 {\times} 10{^{-25}} g cm{^{-3}}. We can
detect stellar PMs corresponding to speeds in the LMC of 770 km s{^{-1}} or
more using the H${\alpha}$ images. Five stars in the remnant show measurable
PMs but none appear to be moving radially from the center, including a
prominent red star 4.6{^{\prime\prime}} from the center. Using coronal [Fe XIV]
{\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
{\pm} 0.41) {\times} 10{^{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 0509-67.5.
