A process for drilling natural gas hydrates. A drilling rig is placed in seawater. A suction-press core drilling mode is adopted in a soft sediment formation, a suction-rotary core drilling mode is adopted in a medium-hard sediment formation, or a pumping direction circle-rotary core drilling mode is adopted in a hard sediment formation. A core is extracted. An inner tube for wireline pressure coring is recovered. A holding seal cap is tightened, and the inner tube is stored in a pipe storage rack. Punching is carried out. An inner tube for wireline pressure coring which is hollow is captured to disengage the holding seal cap. The inner tube is lowered. A drill rod is added. The punching is carried out again. The above steps are repeated until the core drilling reaches a given drilling depth. The drill rod, the drill and the corer are recovered.