We present chemical abundance measurements for 47 damped Lyα (DLA) systems, 30 at z > 4, observed with the Echellette Spectrograph and Imager and the High Resolution Echelle Spectrometer on the Keck telescopes. H I column densities of the DLAs are measured with Voigt profile fits to the Lyα profiles, and we find an increased number of false DLA identifications with Sloan Digital Sky Survey at z > 4 due to the increased density of the Lyα forest. Ionic column densities are determined using the apparent optical depth method, and we combine our new metallicity measurements with 195 from previous surveys to determine the evolution of the cosmic metallicity of neutral gas. We find the metallicity of DLAs decreases with increasing redshift, improving the significance of the trend and extending it to higher redshifts, with a linear fit of –0.22 ± 0.03 dex per unit redshift from z = 0.09-5.06. The metallicity "floor" of ≈1/600 solar continues out to z ~ 5, despite our sensitivity for finding DLAs with much lower metallicities. However, this floor is not statistically different from a steep tail to the distribution. We also find that the intrinsic scatter of metallicity among DLAs of ~0.5 dex continues out to z ~ 5. In addition, the metallicity distribution and the α/Fe ratios of z > 2 DLAs are consistent with being drawn from the same parent population with those of halo stars. It is therefore possible that the halo stars in the Milky Way formed out of gas that commonly exhibits DLA absorption at z > 2.