We present and analyze three-dimensional data cubes of Neptune from the OSIRIS integral-field spectrograph on the 10-m W.M. Keck II telescope, from 26 July 2009. These data have a spatial resolution of 0.035/pixel and spectral resolution of R ~3800 in the H (1.47-1.80 μm) and K (1.97-2.38 μm) broad bands. We focus our analysis on regions of Neptune's atmosphere that are near-infrared dark - that is, free of discrete bright cloud features. We use a forward model coupled to a Markov chain Monte Carlo algorithm to retrieve properties of Neptune's aerosol structure and methane profile above ~4 bar in these near-infrared dark regions.We construct a set of high signal-to-noise spectra spanning a range of viewing geometries to constrain the vertical structure of Neptune's aerosols in a cloud-free latitude band from 2-12°N. We find that Neptune's cloud opacity at these wavelengths is dominated by a compact, optically thick cloud layer with a base near 3 bar. Using the pyDISORT algorithm for the radiative transfer and assuming a Henyey-Greenstein phase function, we observe this cloud to be composed of low albedo (single scattering albedo =0.45-0.01+0.01), forward scattering (asymmetry parameter g=0.50-0.02+0.02) particles, with an assumed characteristic size of ~1μm. Above this cloud, we require an aerosol layer of smaller (~0.1μm) particles forming a vertically extended haze, which reaches from the upper troposphere (. 0.59-0.03+0.04 bar) into the stratosphere. The particles in this haze are brighter (single scattering albedo =0.91-0.05+0.06) and more isotropically scattering (asymmetry parameter g=0.24-0.03+0.02) than those in the deep cloud. When we extend our analysis to 18 cloud-free locations from 20°N to 87°S, we observe that the optical depth in aerosols above 0.5 bar decreases by a factor of 2-3 or more at mid- and high-southern latitudes relative to low latitudes.We also consider Neptune's methane (CH₄) profile, and find that our retrievals indicate a strong preference for a low methane relative humidity at pressures where methane is expected to condense. When we include in our fits a parameter for methane depletion below the CH₄ condensation pressure, our preferred solution at most locations is for a methane relative humidity below 10% near the tropopause in addition to methane depletion down to 2.0-2.5 bar. We tentatively identify a trend of lower CH₄ columns above 2.5 bar at mid- and high-southern latitudes over low latitudes, qualitatively consistent with what is found by Karkoschka and Tomasko (2011), and similar to, but weaker than, the trend observed for Uranus.