The Colours of the Outer Solar System Origins Survey is acquiring near-simultaneous g, r, and J photometry of unprecedented precision with the Gemini North Telescope, targeting nearly 100 trans-Neptunian objects (TNOs) brighter than m r = 23.6 mag discovered in the Outer Solar System Origins Survey. Combining the optical and near-infrared photometry with the well-characterized detection efficiency of the Colours of the Outer Solar System Origins Survey target sample will provide the first flux-limited compositional dynamical map of the outer solar system. In this paper, we describe our observing strategy and detail the data reduction processes we employ, including techniques to mitigate the impact of rotational variability. We present optical and near-infrared colors for 35 TNOs. We find two taxonomic groups for the dynamically excited TNOs, the neutral and red classes, which divide at g − r sime 0.75. Based on simple albedo and orbital distribution assumptions, we find that the neutral class outnumbers the red class, with a ratio of 4:1 and potentially as high as 11:1. Including in our analysis constraints from the cold classical objects, which are known to exhibit unique albedos and r − z colors, we find that within our measurement uncertainty our observations are consistent with the primordial solar system protoplanetesimal disk being neutral class dominated, with two major compositional divisions in grJ color space.