We present rest-frame optical spectroscopic observations of 24 Hot Dust-Obscured Galaxies (Hot DOGs) at redshifts 1.7-4.6 with KECK/NIRES. Our targets are selected based on their extreme red colors to be the highest luminosity sources from the WISE infrared survey. In 20 sources with well-detected emission we fit the key [O III], H$beta$, H$alpha$, [N II], and [S II] diagnostic lines to constrain physical conditions. Of the 17 targets with a clear detection of the [O III]$rm lambda$5007A emission line, 15 display broad blueshifted and asymmetric line profiles, with widths ranging from 1000 to 8000 $rm km s^{-1}$ and blueshifts up to 3000 $rm km s^{-1}$. These kinematics provide strong evidence for the presence of massive ionized outflows of up to $8000 rm M_odot yr^{-1}$, with a median of $150 rm M_odot yr^{-1}$. As many as eight sources show optical emission line ratios consistent with vigorous star formation. Balmer line star-formation rates, uncorrected for reddening, range from 30--1300 $rm M_odot yr^{-1}$, with a median of $50 rm M_odot yr^{-1}$. Estimates of the SFR from SED fitting of mid and far-infrared photometry suggest significantly higher values. We estimate the central black hole masses to be of order $10^{8-10}rm M_odot$, assuming the present-day $rm M_{BH}-sigma_*$ relation. The bolometric luminosities and the estimated masses of the central black holes of these galaxies suggest that many of the AGN-dominated Hot DOGs are accreting at or above their Eddington limit. The combination of ongoing star formation, massive outflows, and high Eddington ratios suggest Hot DOGs are a transitional phase in galaxy evolution.
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