We investigated the electrical characteristics and the charge transport mechanism of pentacene vertical hetero-structures with graphene electrodes. The devices are composed of vertical stacks of silicon, silicon dioxide, graphene, pentacene, and gold. These vertical heterojunctions exhibited distinct transport characteristics depending on the applied bias direction, which originates from different electrode contacts (graphene and gold contacts) to the pentacene layer. These asymmetric contacts cause a current rectification and current modulation induced by the gate field-dependent bias direction. We observed a change in the charge injection barrier during variable-temperature current-voltage characterization, and we also observed that two distinct charge transport channels (thermionic emission and Poole-Frenkel effect) worked in the junctions, which was dependent on the bias magnitude.