© 2015. The American Astronomical Society. All rights reserved. We present new Hubble Space Telescope Cosmic Origins Spectrograph far-ultraviolet (far-UV) spectroscopy and Keck Echellete optical spectroscopy of 11 ultraluminous infrared galaxies (ULIRGs), a rare population of local galaxies experiencing massive gas inflows, extreme starbursts, and prominent outflows. We detect Lyα emission from eight ULIRGs and the companion to IRAS09583+4714. In contrast to the P Cygni profiles often seen in galaxy spectra, the Lyα profiles exhibit prominent, blueshifted emission out to Doppler shifts exceeding -1000 km s-1 in three H ii-dominated and two AGN-dominated ULIRGs. To better understand the role of resonance scattering in shaping the Lyα line profiles, we directly compare them to non-resonant emission lines in optical spectra. We find that the line wings are already present in the intrinsic nebular spectra, and scattering merely enhances the wings relative to the line core. The Lyα attenuation (as measured in the COS aperture) ranges from that of the far-UV continuum to over 100 times more. A simple radiative transfer model suggests the Lyα photons escape through cavities which have low column densities of neutral hydrogen and become optically thin to the Lyman continuum in the most advanced mergers. We show that the properties of the highly blueshifted line wings on the Lyα and optical emission-line profiles are consistent with emission from clumps of gas condensing out of a fast, hot wind. The luminosity of the Lyα emission increases nonlinearly with the ULIRG bolometric luminosity and represents about 0.1-1% of the radiative cooling from the hot winds in the H ii-dominated ULIRGs.