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We present ground-based and Hubble Space Telescope optical and infrared observations of Swift XRF 100316D/SN 2010bh. It is seen that the optical light curves of SN 2010bh evolve at a faster rate than the archetype gamma-ray burst supernova (GRB-SN) 1998bw, but at a similar rate to SN 2006aj, an SN that was spectroscopically linked with XRF 060218, and at a similar rate to the non-GRB associated Type Ic SN 19941. We estimate the rest-frame extinction of this event from our optical data to be E(B - V) = 0.18 ± 0.08 mag. We find the V-band absolute magnitude of SN 2010bh to be Mv = -18.62 ± 0.08, which is the faintest peak V-band magnitude observed to date for spectroscopically confirmed GRB-SNe. When we investigate the origin of the flux at t - t0 = 0.598 days, it is shown that the light is not synchrotron in origin, but is likely coming from the SN shock breakout. We then use our optical and infrared data to create a quasi-bolometric light curve of SN 2010bh, which we model with a simple analytical formula. The results of our modeling imply that SN 2010bh synthesized a nickel mass of MNi ≈ 0.1 M⊙, ejected Mej ≈ 2.2 M⊙, and has an explosion energy of Ek ≈ 1.4 x 1052 erg. Thus, while SN 2010bh is an energetic explosion, the amount of nickel created during the explosion is much less than that of SN 1998bw and only marginally more than SN 1994I. Finally, for a sample of 22 GRB-SNe we check for a correlation between the stretch factors and luminosity factors in the R band and conclude that no statistically significant correlation exists. [ABSTRACT FROM AUTHOR]