Location
The Cave of the Forlorn is found on the 'secret' half of the Mountaintops of the Giants. In order to reach this area you must first retrieve the two pieces of the Haligtree Secret Medallion (one is in the Village of the Albinaurics, while the other is in Castle Sol), then use the completed Medallion at the Grand of Rold. Get through the Hidden Path to the Haligtree and you'll be in a massive snowfield.
Head more or less directly north through the snowfield. Your vision will be limited, thanks to heavy winds, but press onward, minding the occasional big drop. Eventually the whipping snow will ease up, and you'll be on a frozen river that leads east and west. Follow it east and you'll eventually hit a frozen waterfall where Great Wyrm Theodorix is waiting. Before you engage Theodorix, check along the rock walls to your left. You'll see a locked passage guarded by a Giant Land Octopus. Get it out of the way, then use two Stonesword Keys to unlock the passage. The Cave of the Forlorn is within.
(As a side note, it is very easy to beat Theodorix. Simply lure the beast over to the Giant Land Octopuses in this area and let the monsters beat each other to death. You'll receive a ton of Runes and three Dragon Hearts for killing Theodorix, and if you check the frozen waterfall Theodorix was guarding you'll find an Ancient Dragon Smithing Stone.)
Cave of the Forlorn
Activate the Site of Grace, admire the dragon corpse near the entrance, collect the Golden Rune [7] sitting beside the dragon, activate the Summoning Pool, and head inside. In the second half of the entry chamber you'll find four Freezing Greases, and if you look to your right you'll see a winged Misbegotten lurking in the shadows. Take it out before it can ambush you.
There are two paths you can take in the east. If you go left you'll find a smaller, darkened room with a winged Misbegotten and a larger, axe-wielding Misbegotten. There are also three rats hidden in this room, and if you try to go after the Misbegotten they will come out and make the fight miserable. Try to lure the Misbegotten out with ranged attacks before drawing the attention of the rats. Check along the walls of the right side of the room to find a small nook containing a Freezing Grease.
Return to the entry chamber and head east through the larger doorway. Immediately swing left and you'll find a dark side chamber with a winged Misbegotten and an axe-wielding Misbegotten that you can ambush. Pull them back to the entry chamber to fight if you can't get quick kills. They're guarding a Nascent Butterfly, and another chamber just south of this one contains a Golden Rune [10].
Carry on east until you reach another larger chamber. Despite how it appears this area is completely safe. Climb the pile of rock and snow dead ahead to get at a Hero's Rune [2]. There are plenty of Miquella's Lillies in this chamber if you want to craft Bewitching Branches, as well. To the north is a darkened passage, and you'll hear a sleeping Misbegotten. If you're quiet you can slip right past the thing and find the next larger chamber by going left.
There are three more sleeping Misbegotten in the next area, and it's unlikely that you'll sneak past them undetected. The larger, axe-wielding Misbegotten nearest you has trouble getting up the ramp near the entrance, so you can somewhat safely snipe the winged Misbegotten further in from higher ground. The third Misbegotten also has an axe, and can be dispatched in the same fashion as the first. In the rear of their chamber you'll find ten Spiritflame Arrows and a Rune Arc.
At the end of the path you'll find three more Thawfrost Boluses, along with a hole guarded by three more Spirit Jellyfish. This bit's a little trickier since they're all in such close proximity to you, so try to draw at least one of them back the way you came. Drop carefully into the hole and you'll find five Freezing Greases as you make your way down the platforms. At the bottom is boss fog.
Misbegotten Crusader
Well isn't that a fancy sword. The Misbegotten Crusader is, despite its higher health and damage output, for the most part functionally identical to the Leonine Misbegotten that you likely ran into a few times earlier in Elden Ring. The only major difference here is that it can use Establish Order, resulting in an AOE shockwave and a wide projectile attack aimed straight at you.
Your greatest enemy in this battle is the boss chamber, as it's not huge. It's pretty easy to get backed into corners as you try to avoid the Misbegotten's attacks. Watch its sword swipes carefully and wait for the Misbegotten to hold off on its hits before getting in your own attacks. The moment the Misbegotten pulls a pose and begins to glow you need to get clear, then prepare to avoid an incoming projectile. Establish Order can quickly wipe out Spirit Ashes, so while summons are great in this fight if you can keep the Misbegotten occupied you should tend towards ranged attackers rather than melee spirits.
You'll earn 93,000 Runes and the Golden Order Greatsword for defeating the Misbegotten Crusader. You'll also complete the Cave of the Forlorn. (Though when you return to the surface you'll find a Giant Land Octopus blocking your path back out of the cave, so... maybe consider warping away rather than walking.)