We report the fabrication of both n-type and p-type WSe2 field effect transistors with hexagonal boron nitride passivated channels and ionic-liquid (IL)-gated graphene contacts. Our transport measurements reveal intrinsic channel properties including a metal-insulator transition at a characteristic conductivity close to the quantum conductance e2/h, a high ON/OFF ratio of >107 at 170 K, and large electron and hole mobility of ~200 cm2V-1s-1 at 160 K. Decreasing the temperature to 77 K increases mobility of electrons to ~330 cm2V-1s-1 and that of holes to ~270 cm2V-1s-1. We attribute our ability to observe the intrinsic, phonon limited conduction in both the electron and hole channels to the drastic reduction of the Schottky barriers between the channel and the graphene contact electrodes using IL gating. We elucidate this process by studying a Schottky diode consisting of a single graphene/WSe2 Schottky junction. Our results indicate the possibility to utilize chemically or electrostatically highly doped graphene for versatile, flexible and transparent low-resistance Ohmic contacts to a wide range of quasi-2D semiconductors. KEYWORDS: MoS2, WSe2, field-effect transistors, graphene, Schottky barrier, ionic-liquid gate