Reaching Archdragon Peak
To initially reach Archdragon Peak, you will need to defeat Oceiros, the Consumed King and explore the passage behind his boss arena to find the Path of the Dragon gesture. Using the gesture here in Irithyll Dungeon will teleport you to the initial bonfire in Archdragon Peak.
Upon reaching Archdragon Peak, you will make an ascent to the first bonfire. Don't worry if you die, you will only be sent back to the spot you teleported in at. Head forward and pick up the Soul of a Crestfallen Knight. Continue forward, watching out for the Serpent-men ahead. At the fork in the path, take the left path and pick up 2 Homeward Bones.
You will see a lift in front of you. This is a shortcut to this area from the Dragon-Kin Mausoleum bonfire, so you cannot use it yet. Push foward, killing the three Serpent-men and pick up the Titanite Chunk amongst the rocks. You have now reached the Archdragon Peak bonfire, rest at it and look to your immediate right to pick up an Ember.
Having rested, continue up the staircase and pick up the Soul of a Weary Warrior to your right, and the Titanite Chunk to your left. Continue following the left path up some stairs and deal with the Serpent-men and Large Serpent-men. Nearby pick up the Ancient Dragon Greatshield. Head down the stairs, picking up the Titanite Chunk by the side of it, and before pulling the lever, head to the left down a sloped path, where you will find the Lightning Clutch Ring guarded by a Serpent-man. Return to the lever and pull it to enter the Ancient Wyvern boss fight.
The Ancient Wyvern
The actual boss fight here is very easy. Simply head past the Ancient Wyvern and to the left, traversing through the ruins, across the gatehouse roof and onto some scaffolding. From here, roll on to a stone ruin next the the Ancient Wyvern's head and plunging attack it to defeat it. I recommend doing this first before trying to collect the items. As you are teleported after the fight, return by either using the shortcut you saw ealier from Dragon-kin Mausoleum or through the passage between the Dragon-kin Mausoleum and the top of the gatehouse. The Ancient Wyvern will drop the Dragon Head Stone.
These directions will assume you are starting from entrance for the Ancient Wyvern. Head forward up the staircase, and pick up the Large Soul of a Crestfallen Knight in the centre of the platform. Head to the right, to find three corpses, two holding an Ember, and the other holding 6 Stalk Dung Pie. Before heading back, walk around the ruined balustrade, so that you are facing the pillars. Continue forward to pick up the Ring of Steel Protection. Return to the central platform.
Now take the left route from the central platform, picking up the Large Soul of a Weary Warrior. Walk up the stairs and head to the left to pick up the 4 Lightning Urns. Continue through the ruined template, dealing with the Serpent-men. Cross the stone walkway into another ruined temple, where one of the circular walls has collapsed. Deal with the Serpent-man and pickup the Titanite Chunk near to him. Opposite this, you will see a ladder, climb down it to 2 Twinkling Titanite. Climb back up and proceed forward, picking up the 3 Dung Pie just before the staircase.
Ascend the staircase, watching out for the two Serpent-men who drop down behind you. Proceed upwards, and once at the top deal with the Serpent-men. Head through the passage to the left, following it around to the top of the ruined temple. Deal with the Serpent-men and Large Serpent-men. Here you will find a corpse holding 2 Titanite Chunks, another holding 12 Lightning Bolts.
Head back to the walkway and proceed forward, dealing with the Serpent-men. Take caution with the chained axe Large Serpent-man as his range is extremely long. Once across the walkway, climb up the ladder onto the scaffolding. Proceed forward all the way along the scaffolding to pick up 2 Twinkling Titanite.
Return to the Dragon-kin Mausoleum bonfire.
The Dragon-Kin Mausoleum
Before attempting to do anything else, deal with the Serpent-Man Summoner and his summoned spirit. Having done so, pick up the Soul of a Crestfallen Knight found on the upper level. On the upper level, head to the right and through a doorway onto a balcony. Deal with the two Serpent-men and pick up the Titanite Scale next to them. After this, head back into the mausoleum and activate the shortcut lift. After doing that, enter the balcony on the left-hand side of the mausoleum that overlooks the Ancient Wyvern arena. Pick up the Titanite Scale on the corpse there.
Proceed forward, watching out for the Rock Lizard. You will return to the lower floor of the mausoleum. Now head straight up and towards the porticullis to pick up the Dragonslayer Spear. Head to the right and down a stairway into a passage with several Serpent-Men. Deal with them and proceed back outside, where you will see a Large Serpent-men. Deal with him, but watch out for the hidden Serpent-man on a lower platform opposite to him.
Having dealt with the immediate threats, walk towards the pillar closest to the wooden platform. There is a ladder, ascend it to find a corpse holding the Thunder Stoneplate Ring. From here, walk down the stone staircase onto the wooden platforms and proceed forward across the makeshift wooden bridge. Upon entering the temple after the bridge, you will see a Crystal Lizard. Kill it to get 3 Twinkling Titanite. Head to the left to find a wooden chest containing 3 Titanite Scale. Backtrack and take the left path back outside. You will encounter several Rock Lizards here, as well as another Ancient Wyvern. If you kill the Ancient Wyvern, it will drop 6 Titanite Chunks, 3 Titanite Scales and 3 Twinkling Titanite. You don't actually have to fight the wyvern, if you run ahead it will fly off.
Having dealt with the other Ancient Wyvern in either fashion, pick up the Soul of a Weary Warrior on the corpse in the window looking into the ruined temple. Then continue forward into another ruined temple complex, this time containing a Serpent-man Summoner who summons an enemy that, when killed for the first time, will drop Ricard's Rapier. Continue forward to reach the Great Belfry bonfire.
The Great Belfry
This bonfire is right outside the great bell used to summon the Nameless King to fight. To do so, head back towards the Serpent-Man Summoner into the open-air passage, but head to the right towards a lever in front of the giant bell. Ringing this bell will make Archdragon Peak stormy, chaning the visual effects and ambient sounds until you defeat the Nameless King. I recommend exploring the rest of the Great Belfry before taking him on.
Proceed straight forward out into a courtyard with several Serpent-Men. Once you start fighting them, a gauntlet will start with a few more Serpent-men appearing and three Large Serpent-men accompanying them. If you survive this onslaught, you can find the Dragon Chaser's Ashes on the small secluded path to the left of the ruined building. Watch out for the Rock Lizard. If you continue forward you will find a dragon altar. Use the Path of the Dragon gesture in front of the altar to gain the Twinkling Dragon Torso Stone.
Head backwards, enter the lower courtyard through the two open portcullis. Before doing so, pick up the Large Soul of a Crestfallen Knight on a corpse nearby. Once in the lower courtyard, pick up the Ember on the corpse near to the closed portcullis. You cannot proceed here until you defeat the Nameless King, so backtrack and take the stairs to the right. You will enter a balcony with a Rock Lizard, a wooden chest and a ladder. Open the chest to get 3 Twinkling Titanite.
Proceed up the ladder to see a dead Ancient Wyvern with Havel the Rock standing in front of him, guarding a Titanite Slab. Pick up the Titanite Slab and kill him to get Dragon Tooth and Havel's Greatshield. Killing him here will cause Havel's Set to spawn where the Stray Demon was in Farron Keep, so remember to return there to pick it up. Having defeated Havel the Rock, drop down on to the ruins to the left of the dead Ancient Wyvern. Proceed forward to reach a Rock Lizard guarding the Great Magic Barrier miracle.
Now it is time to face the Nameless King.
The Nameless King
Return to the lever in front of the giant bell, and pull it. The weather will change and a walkable fog will appear underneath the bell. Drop down on it and enter the boss fog to start the Nameless King boss fight.
During the first phase of the fight, you will fight the King of the Storm, his wyvern. It is generally easier to damage him with ranged attacks than melee. Be wary of sticking underneath him, as he will lift off and breathe fire directly downwards as a counter to it. The Nameless King himself will occasionally throw lightning bolts or attack with his Dragonslayer Swordspear. Dodging these is fairly easy. The biggest difficulty here is the camera, as the King of the Storm flies around a lot, which makes tracking him and the Nameless King hard at times.
Once the King of the Storm is dead, the Nameless King will attack you in hand-to-hand combat. Many of his attacks have a delayed quality to them, throwing off your dodge timing if you are not careful. He hits quite hard, and will punish you easily if you try and estus too at the wrong moment. Watch out for his dash attack, where he will dash to the side and thrust with his swordspear. Generally, a mix of forward and side rolls will avoid most of his attacks. Back rolls are liable to be punished by his long reach.
At a distance, he will do dash and thrust attacks, with a long-range. as well as a magic attack where he charges his swordspear and launches a lightning bolt at your location. The tell for this attack is very noticeable, so just dodge to the side when it occurs. For more detail on the fight, refer to his boss page.
Once the Nameless King is defeated, the storm will subside and you will appear in the courtyard beneath what was his arena. The portcullis that was once closed is now open, allowing access to the Great Belfry from the Dragon-Kin Mausoleum. From the Nameless King bonfire, head forward and pick up the Titanite Slab on a corpse in the central path. Proceeding down the central path you will find another corpse to the left, holding the Dragonslayer Set.