The Treacherous Path to Purgatory
Carefully cross the bridge and wait for the enemies to notice. You have to watch out for the big and fast enemy that can cause extensive bleeding damage with its weapon. If possible, lure the undead and take them out before dealing with him. Continue uphill and loot the corpse hanging by the cliff to get a Notched Whip.
Backtrack a bit then continue uphill along the narrow path. There more of those fast enemies on top of the pillars so don't carelessly run towards them and aggro them all. Cross the hanging bridge while carefully minding the holes along the way. You'll find a dark spirit invader right before the misty door; defeat it then loot the corpse sitting by the wall beside the bridge to get a Sublime Bone Dust and Fire Seed. You can immediately return Majula and burn it there to upgrade the healing effectiveness of your Estus Flask. There's a boss past that misty entrance as well. Refer to the Undead Purgatory section for more details.
Valley of Rogues and Skeletons
Backtrack to the large iron bridge you lowered earlier and head to the third bridge. There's a bonfire inside a locked room there. Ignore it for now. Before crossing the bridge, enter the tunnel to the right. There are skeletons that will be revived over and over again. Loot the first corpse to get a Magic Mace and Titanite Shard. Head inside and find the necromancer to stop the skeletons from getting revived.
Light up the torches inside if you want but you won't be able to open the gate from that side. Find the other path leading outside; just be careful of the rogues and archers along the narrow pathway along the cliff. From there, follow the path to the left (which leads back to the third bridge with the third bonfire) and take out any rogues. There should be an uphill path to the right.
Follow it and you'll find a corpse in the middle. Don't pick it up yet; there are hiding rogues to the right side; deal with them first. Loot the Poison Throwing Knife x10 from the corpse then carefully peek over the edge to find a small ledge you can drop on with a lootable corpse. Get Poison Moss x3 from it then drop to the main path below.
There are two enemies waiting inside the room to the side so make sure you don't get taken by surprise. Loot the Undead Lockaway Key. This is the key needed to unlock the door of the third bonfire. Backtrack and light it up. It's up to you if you need to rest and refresh. You can also talk to Creighton inside and exhaust his dialogue options. You can learn the Fist Pump gesture from him as well.
Backtrack to the cliffside and follow the path again leading to the waterfall. On your way there, a dark spirit invader will appear. It has really damaging attacks and can easily break your guard so push it off the cliff to easily defeat it.
Once you've reached the ruins, take the path by the waterfall and you'll find a misty door. Don't enter it yet; inside, follow the other path that leads to the skeleton cave. The necromancer is located on a separate ledge to the left but before dealing with it, lure the other skeletons and take them out. They won't resurrect immediately so use this chance to deal with the necromancer.
Once clear, take the path to the right that leads to a broken hanging bridge with a corpse on it. Loot the Combustion spell and Titanite x3 from it. Continue forth and take out one last skeleton. Loot the corpse behind it to get Flame Butterfly x3. Pull the lever there to open the gate and create a shortcut for you leading to the third bonfire. Now get ready and enter the misty door.
Boss: Skeleton Lords
There are three Skeleton Lords and they all share the same HP bar. As soon as you enter, run to the right and take out the skeleton mage while they're still not ready. After defeating it, wait a bit and defeat the skeleton warriors that will spawn. The other two skeleton warriors have wide range and can use fireballs as ranged attacks but they're slow so its possible for you to outrun them. Run around and use the pillars as cover whenever you need to heal.
Once the spawns are defeated, take out the next skeleton lord and the warriors it will spawn upon defeat. Take out the last skeleton lord and make sure to defeat the skeleton wheel spawns before they get to move. They're fast and can be problematic if they hit you all at once.
Now continue along the path and pull the lever to lower the bridge. Continue along the path to reach the next area. From here, you can go ahead and explore Harvest Valley or return to Huntsman Copse to clear Undead Purgatory.