In this paper and in a companion paper by Wise, McNamara, & Murray, we present a detailed, multiwavelength study of the A1068 galaxy cluster, and we use this data to test cooling and energy feedback models of galaxy clusters. Near-ultraviolet and infrared images of the cluster show that the cD galaxy is experiencing star formation at a rate of ~20-70Msolar yr-1 over the past <~100 Myr. The dusty starburst is concentrated toward the nucleus of the cD galaxy and in filamentary structures projecting 60 kpc into its halo. The Chandra X-ray image presented in Wise, McNamara, & Murray reveals a steep temperature gradient that drops from roughly 4.8 keV beyond 120 kpc to roughly 2.3 keV in the inner 10 kpc of the galaxy, where the starburst peaks. Over 95% of the ultraviolet and Hα photons associated with the starburst are emerging from regions cloaked in keV gas with very short cooling times (~100 Myr), as would be expected from star formation fueled by cooling condensations in the intracluster medium. However, the Chandra spectrum is consistent with but does not require cooling at a rate of 114-145Msolar yr-1, factors of several below the rates found with the ROSAT observatory. The local cooling rate in the vicinity of the central starburst is <~40Msolar yr-1, which is consistent with the star formation rate determined with U-band and infrared data. We consider energy feedback into the intracluster medium by the radio source, heat conduction, and supernova explosions associated with the starburst. We find that energy feedback from both the radio source and thermal conduction are inconsequential in A1068. Although supernova explosions associated with the starburst may be able to retard cooling in the inner 10 kpc of the cluster by ~18% or so, they are incapable of maintaining the cooling gas at keV temperatures. Therefore, it is plausible that the star formation in A1068 is being fueled by the cooling intracluster medium. Finally, we present circumstantial evidence for the contrary view that at least some and perhaps all of the star formation may have been fueled by an interaction between the cD and one or more companion galaxies.