Bros, do I have to kill Vendrick to obtain the fourth crown?
Yeah, you need to collect Vendrick's crown, which involves killing his hollowed out form and the travelling to the Shrine of Amana to open the locked door past the "hidden" (underwater) bridge. I believe you also need to be human in order to open this and collect his soul, armor, and crown.
Fighting Vendrick isn't particularly pleasant. It's one of those bosses designed more for atmosphere and "demFeelz" as opposed to fun. Vendrick's anemic looking swings deal enormous amounts of damage which I surmise is meant to showcase the hollowed king's power and make the player question just how powerful he was in his prime. Unfortunately, while this may make for an interesting piece of lore to sperg about or make videos earning you six figures on Patreon, it makes for a stultifying boss encounter. You want to bring along at least 4 giant souls, if not a full 5 in NG+ and beyond. Once in the fight, hug his left (?) buttcheek and attack during the recovery period of his swings. Try not to let him get backed up against a wall either. You'll want to equip yourself either for lots of damage (Ring of Blades, Flynn, Old Leo) or to avoid being one-shotted (Ring of Steel Protection +2, decent armor, Life Ring +3, Third Dragon Ring). If you have access to the Denial miracle, you might want to use that as insurance.