Fighting the Moon Presence can only happen in the game's third (some say "true") ending, achieved only by consuming three One Third of the Umbilical Cord items and defeating Gehrman, the First Hunter. Do this, and the game's ultimate boss will be ready for battle.
Despite being the game's final boss, the Presence is actually not too difficult. It moves fast, but leaves itself open for plenty of regain attacks. At close range, it'll attack with grabs and swipes - sidestep around and attack his tails. At a medium range it'll lunge, but you can roll under it if you move quickly.
What you need to watch for is its unavoidable red energy blast - this move reduces your HP to 1, every time. Fortunately, it stands motionless for a while after, so use the time to heal or regain health with some attacks. Finally, the Presence may summons floating orbs. These burst and drop blood onto the battlefield, so be sure to avoid the splash. If you're hit, you won't be able to heal for a limited time.