We present results from high-resolution spectra of 237 stars in the Orion Nebula cluster (ONC) obtained during two engineering runs with the Hectochelle multifiber echelle spectrograph on the 6.5 m MMT. The ONC is the nearest populous young (age ~ 1 Myr) cluster and is therefore an important object for studies of the evolution of protoplanetary disks. Using the high spectral resolution of Hectochelle, we are able to distinguish stellar accretion and wind emission-line profiles from nebular emission lines and identify accreting members of the cluster from Hα profiles with greater accuracy than previously possible. We find 15 new members on the basis of Li λ6707 absorption and Hα emission. Line profiles of Hα of some objects that are not too contaminated by nebular emission show features characteristic of mass inflow and ejection. We also present rotational velocities as part of an initial investigation into angular momentum evolution of very young stars, confirming a difference between classical T Tauri stars and weak-line T Tauri stars that had been found from period analysis. Finally, we present an initial study of the radial velocity dispersion of the brighter stars in the central cluster. The very small dispersion derived, ≤1.8 km s-1, is in good agreement with estimates from proper motions.