Start off by examining the nearby chest, then take the stairs to get to where the screen shows. Star wall-running, but make sure to bounce off twice towards the opposite wall so as to avoid falling down into the lava. Run to the interactive edges and grab onto them.
Use the edges to get to the nearby balcony. Take the only available path, but watch out for more Legion Soldiers. Your destination is the shaft pictured above - use Death Grip on the handhold there.
DON'T do any climbing, instead let go of the first edge Death reaches. The trick is to drop down to the bottom edge shown on the screen, which the character should automatically grab.
Now, move to the right as far as possible and climb a new shelf. Go through another corridor and stop when you see a new handhold on the right (the above screen).
Star climbing (screenshot 1) by grabbing onto the handholds with Death Grip and bouncing off the walls. You should reach a small area guarded by a Legion Champion (screenshot 2). Take him down, but be more careful this time because the smaller size of the fighting area will make it more difficult for you to dodge. After you've won, look around for a Boatman Coin and a legendary primary weapon - Demonflame Renders.
Bo back to the shaft, drop down and resume the run to find a new interactive handhold seen above. Grab onto it with Death Grip.
Run on the wall vertically and bounce off it (screenshot 1). Keep bouncing off like that until you reach a small shelf with a chest you've passed by earlier (screenshot 2). Look inside to find a Dungeon Map.
Repeat all the steps connected to getting to the first large shaft, so wall-run on the walls and run through a few corridors. Use Death Grip on the handhold again, but this time don't let go of the edge, but start going up (the above screen), grabbing onto a handhold and a new edge.
Finally, move far to the right and climb the new shelf. Before leaving this place, look around for a Book of the Dead page ([The Book of the Dead] side quest). Use the nearby door to go back to the central area of the Black Stone and interact with the switch shown above.
Be cautious, because hitting the switch will cause a Legion Champion to show up somewhere nearby. Consider retreating to the main arena so that you could have a wider battlefield while fighting the demonic beast. After winning the battle, examine the area north of the switch and you'll find the interactive edges pictured above.
After grabbing onto the first edge, move left and start lowering yourself down so as to land in the bottom corridor. Deal with the Legion Soldier there, then look inside the nearby chest.
Now it's time to go back to the balcony with the switch, so grab onto the interactive handhold, and then an edge. Once at the north-eastern balcony, select Phasewalker from the ability list and activate the green portal shown on the above screen. Go through it to go back to the present.
Start exploring the area by checking out the chest and picking up a Soul Arbiter's Scroll. Afterwards, go to the central arena seen on the screen. There is a fresh batch of Undead Scarabs to eliminate there.
Once they're dealt with, stand in front of the handhold, as seen on the screen, and use Death Grip on it. Wait until Death gets to the upper balcony and use the door leading to the western part of the Black Stone.