The Old Demon King is the optional boss in the optional area that is the Smouldering Lake, below the Catacombs of Carthus. The Demon King makes a lot of noise and talks a big game, but overall this guy's bark is worse than his bite if you're prepared for it, and you can get a tasty boss soul out of it too.
Getting to the Demon King's arena is really easy, especially when compared to some of the other bosses. Start at the Demon Ruins bonfire by the Smouldering Lake and run back up the stairs and out. The door to the Old Demon King will be on your right. The only thing you need to watch out for is the giant ballista, if you haven't yet turned it off, but even if you haven't it's easily avoided at this range.
The Old Demon King uses fire, so hurling pyromancies at him is going to do next to nothing. Pyromancies that boost your own resistance to fire, however, are a good idea, as are any other flame resistant items you may have around. The Old Demon King is weak to dark,so if you can buff your weapon with that, or have dark sorceries and pyromancies, then so much the better.
This is another fight that's easier for melee players. If you get in close to him and get tucked under his belly in front of him, you're going to avoid taking damage from most of his melee attacks, although you'll only get a few hits in there before he'll notice you're there and turns his club to slam the heavy end down vertically, which does area damage, so roll away when you see him telegraphing that. If you get in behind him you'll likewise be able to get hits in, but he'll eventually turn and slam his club down where you were standing.
The ranged attacks he commonly uses in this phase are a big fireball/explosion of fire and a long fire breathing attack that he sweeps across the arena in front of him. The range on both of these is fair, but avoidable, and both can be avoided if you're close enough to him to be underneath his head rather than in front of it. In the case of the fire breathing, it actually renders him vulnerable to attack and is also easily avoided by running to his left (your right) when he dusts it off. The more difficult fire attack to avoid for melee fighters is an area of effect one he conjures by dripping fire from his mouth, which sends flames shooting out in lines. This is easy to get away from at range, but close up it's much harder.
For ranged and spellcaster types you can use the heaps of dead, calcified abyss demons to move behind and around for cover. His ranged attacks can pass through them, but will often pass over your head rather than connect, and you'll give yourself enough cover to use any estus flasks you need. Running around and between them should also get you enough breathing space to cast your spells or fire your arrows.
After you knock off half his health the Old Demon King goes flame on, so his melee attacks now do more flame damage provided they hit you. He's also a little more aggressive and smashes around a bit more. The Demon King also adds two more attacks to his repertoire: he taps his club twice on the ground, slowly, which generates an expanding circle of flames, and he can conjure a bunch of small floating meteors which will then slowly home in on you en masse. These are actually both easy to avoid, since the meteors are extremely slow and easy to outrun so they smash into either the ground or the heaps of dead demons, and the flame circle will burn itself out at mid range from the Old Demon King, though if you run too far away from him as he's conjuring this one the flame circle will start behind you and travel towards him, because otherwise it would be too easy.
You can apply the same tactics you did in the first phase as long as you keep these two attacks in mind. They're telegraphed obviously so you should have enough warning to prepare for them. The Old Demon King has one last dastardly trick up his sleeve, though. Once he gets down to minimal health he'll collapse on the ground, and when he does this you need to immediately back away from him. He will generate a big explosion that does a lot of damage in a last ditch attempt to kill you, so if your health is even questionable it might succeed, undoing all your good work until this point. After this, though, it's easy to move in and finish him off.
Buy the official Dark Souls 3 Guide now.