We are carrying out a large ancillary program with the SDSS-III, using the fiber-fed multi-object NIR APOGEE spectrograph, to obtain high-resolution H-band spectra of more than 1200 M dwarfs. These observations are used to measure spectroscopic rotational velocities, radial velocities, physical stellar parameters, and variability of the target stars. Here, we describe the target selection for this survey and results from the first year of scientific observations based on spectra that is publicly available in the SDSS-III DR10 data release. As part of this paper we present RVs and vsini of over 200 M dwarfs, with a vsini precision of ~2 km/s and a measurement floor at vsini = 4 km/s. This survey significantly increases the number of M dwarfs studied for vsini and RV variability (at ~100-200 m/s), and will advance the target selection for planned RV and photometric searches for low mass exoplanets around M dwarfs, such as HPF, CARMENES, and TESS. Multiple epochs of radial velocity observations enable us to identify short period binaries, and AO imaging of a subset of stars enables the detection of possible stellar companions at larger separations. The high-resolution H-band APOGEE spectra provide the opportunity to measure physical stellar parameters such as effective temperatures and metallicities for many of these stars. At the culmination of this survey, we will have obtained multi-epoch spectra and RVs for over 1400 stars spanning spectral types of M0-L0, providing the largest set of NIR M dwarf spectra at high resolution, and more than doubling the number of known spectroscopic vsini values for M dwarfs. Furthermore, by modeling telluric lines to correct for small instrumental radial velocity shifts, we hope to achieve a relative velocity precision floor of 50 m/s for bright M dwarfs. We present preliminary results of this telluric modeling technique in this paper.