Birthright
Receive this quest when you talk to your mother at Gamlen's House. Then talk to your sibling (Carver/Bethany) and agree to go search for the will. Head to Darktown, and there to Amell's estate cellar.
Enter the cellar. In the room to the north, find a chest that contains either Tiberius's Documents or Portrait of your Mother. This begins a companion quest, either Family History or Portrait of the Past depending on whether Carver or Bethany is with you. Head west, killing enemies along the way. One of them should have a key to the vault room. Once you loot it, head up to the vault room and get the will from one of the chests. If you are done with the area, agree to Carver's suggestion to leave the area. This concludes the quest.
Family History
This quest is available only if Carver is the surviving sibling, and it starts after finding Tiberius's Documents in Amell's estate cellar. After you are done with Birthright, head to the Gallows and speak to Tiberius. He will give you Father's Letters. Then return to Gamlen's home and talk to Carver and show him the letters. This should give a big boost to your friendship with him.
Portrait of the Past
This quest is available only if Bethany is the surviving sibling, and it starts after finding the Portrait of your Mother in Amell's estate cellar. After you are done with Birthright, talk to Bethany at Gamlen's house and offer the portrait to her as a gift. This will give a big boost to your friendship with her.
The Bone Pit
This quest starts after reading a letter on your writing desk, after you have completed the Birthright quest. Head over to Hubert in the Hightown Market. After talking to him, exit the city (by leaving the area and selecting the icon on the bottom right corner of the map) and go to the Bone Pit. There is a skeleton just south of the entry area; make sure to interact with it to receive the "Beared Beast" quest. Follow the path and find the Bone Pit entrance. Venture through the mines, heading south. You will encounter a surviving miner, talk to him then enter the Bone Pit Ledge. Get ready for a difficult fight against a Mature Dragon. Then return to speak with Hubert in Higdhtown.
Get Back to Work
After completing The Bone Pit quest, agree to Hurbert's request to convince the miners to get back to work. Head to Lowtown and talk to the miners near The Hanged Man. Talk to Jansen and convince him to go back to work at the mine.
Remains of the Outlaw "Bearded Beast"
You receive this quest when interacting with the skeleton in the Bone Pit area. Deliver Arintal's remains to Arintal the Lesser in the Docks.
Eurice's Pommel
This quest is started when you find Eurice's Pommel in the Bone Pit Mines. Head to Viscount's Keep Barracks, and deliver them to Guardsman Eustice who is behind a door in the SW area of the barracks.
