Food processor, high-powered blender, or something else capable of producing apples puree. Cheesecloth, preferably lots of it (though it's possible that a sieve would also work- or maybe coffee filters?). Pitcher/bowl/container for juice. And something else to put the dry pulp into.

Place baking tray on base with pierced hole hanging over cider collection jug. Fill with apple pulp midway up the bucket. Place wooden press top on wood pulp and Jack on top so that the jack will press down on the press and up against the top of the frame when pumped. Pump up the jack and press out the good stuff.