To reach Archdragon Peak you need to travel to Irithyll Dungeon, cross the bridge to the opposite side of the cell block, and then turn left and drop to the first floor to take the lift at the end of the hall.
This leads to the area with the dragon-like statue meditating and you can now perform the Path of the Dragon gesture to teleport to Archdragon Peak.
Follow the path and grab the Soul of a Weary Warrior near the mangled tree to the left and then head up the hill to encounter two Serpent-Men.
These enemies use a variety of weapons that you can easily block with a shield, but they also have the ability to extend their necks to bite at you. Back up when you notice that they are stopping to extend their necks, as these attacks take a large chunk of stamina, and make sure that they aren’t blocking as they can parry your attacks.
Head straight to the tip of the stone outcropping for a Lightning Gem and then backtrack a bit and turn right to find the Homeward Bone x2 on the corpse leaning over the cliff edge.
Up ahead are two Serpent-men that spew fireballs at you for ranged damage and there are two melee version hiding near them. Use ranged attacks to deal with the fireball spewing versions before you head up to deal with the remaining ones, collect the Titanite Chunk , and activate the bonfire next to an Ember.
Take the stairs and turn left before the arch to find a Titanite Chunk and then take the stairs up to the top of the gate. There is a Serpent-man closest to you and behind him is a Large Serpent-man.
The larger versions use a greataxe that it sweeps around in heavy attacks that are fairly easy to predict and avoid when you are close, but at long range the head of the axe can be thrown like a ball and chain.
There is an Ancient Dragon Greatshield on the tower section with the Large Serpent-man and a Titanite Chunk on the ground near the opposite stairs.
Take the path to the left of the lever to encounter another Serpent-man and then collect the Lightning Clutch Ring . You can now pull the lever and move forward until the Ancient Wyvern swoops down.