We present mid-IR spectral decomposition of a sample of 48 Spitzer-selected ULIRGs spanning z ~ 1-3 and likely L_IR ~ 10¹²-10¹³ L. Our study aims at quantifying the star formation and active galactic nucleus (AGN) processes in these sources that recent results suggest have evolved strongly between the observed epoch and today. To do this, we study the mid-IR contribution of PAH emission, continuum, and extinction. About of our sample are continuum (i.e., AGN) dominated sources, but ~60% of these show PAH emission, suggesting the presence of star formation activity. These sources have redder mid-IR colors than typical optically selected quasars. About 25% of our sample have strong PAH emission, but none are likely to be pure starbursts as reflected in their relatively high 5 μm hot dust continua. However, their steep 30-14 μm slopes suggest that star formation might dominate the total infrared luminosity. Six of our z ~ 2 sources have EW_6.2 0.3 μm and L_{14 μm} 10¹² L (implying L_IR 10¹³ L). At these luminosities, such high EW_6.2 ULIRGs do not exist in the local universe. We find a median optical depth at 9.7 μm of τ_{9.7 μm} = 1.4. This is consistent with local IRAS-selected ULIRGs but differs from early results on SCUBA-selected z ~ 2 ULIRGs. Similar to local ULIRGs, about 25% of our sample shows extreme obscuration (τ_{9.7 μm} 3), suggesting buried nuclei. In general, we find that our sources are similar to local ULIRGs but are an order of magnitude more luminous. It is not clear whether our z ~ 2 ULIRGs are simply scaled-up versions of local ULIRGs or subject to fundamentally different physical processes.