Sun-like stars are thought to accrete most of their final mass during the protostellar phase, during which the stellar embryo is surrounded by an infalling dense envelope. We present an analysis of 26 K-band spectra of Class 0 protostars, which are the youngest protostars. Of these, 18 are new observations made with the Keck MOSFIRE instrument. H i Brγ, several H2, and CO Δv = 2 features are detected and analyzed. We detect Brγ emission in 62%, CO overtone emission in 50%, and H2 emission in 90% of sources. The H i and CO emission is associated with accretion, while the H2 lines are consistent with shock excitation indicating jets/outflows. Six objects exhibit photospheric absorption features, with almost no outflow activity and no detection of the accretion-related Brγ emission line. Comparing these results with an archival sample of Class I K-band spectra, we find that the CO and Brγ emission lines are systematically more luminous in Class 0s, suggesting that the accretion is on average more vigorous in the Class 0 phase. Typically associated with the heated inner accretion disk, the much higher detection rate of CO overtone emission in Class 0s indicates also that episodes of high accretion activity are more frequent in Class 0 systems. The kinematics of the Class 0 CO overtone emission suggest either an accretion-heated inner disk or material directly infalling onto the central region. This could point toward an accretion mechanism of different nature in Class 0 systems than the typical picture of magnetospheric accretion.