Birthright is a companion quest in Dragon Age II. During this quest you'll be searching for your Grandfathers will, which could prove that the Amell estate belongs to your family.
Acquisition
Cellar Entrance in Darktown
In Act 1, stop in at Gamlen's house to hear Leandra and Gamlen argue about money and about Grandfather Amell's will. Gamlen will say that nothing was left for Leandra. He will also say that the will is still in the Amell estate vault, which is no longer accessible since slavers now own the estate. After this conversation, speak to Carver or Bethany and discuss how to break into the vault in order to initiate the quest
Walkthrough
The underground passage into Amell estate cellar can be found outside of Anders's Clinic in northern part of Darktown. Once inside the passage, expect some resistance from various slaver guards. Loot of various sorts can be found in the side rooms. In the storeroom you can find a gift for Hawke's sibling (for the quest Portrait of the Past or Family History) and a codex entry, Codex entry: The Amell Family.
Chest containing item for the sibling quest (Bethany or Carver) and the Amell Family Crest.
In the side room just before the Anteroom you can find more loot and another codex entry, Codex entry: History of Kirkwall: Chapter 2.
In the final room before the vault, the party will face a slaver mage as well as various other enemies. Disable the trap (Simple) in the room with a rogue to make the fight easier. The key to the vault can be found on the mage's body.
Enter the vault to pick up various loot as well as Grandfather Amell's will. The chest containing the will also holds an armor upgrade for Carver/Bethany. You will gain 
(+15) approval from Bethany upon entering the vault.
Chest in the family vault containing the Leandra's parent's will.
Heirloom Amell Protective Sigils, magic resistance +10% for Bethany
Fereldan Girded Plating, armor +10% for Carver
Another codex entry can also be obtained, Codex entry: History of the Circle.
At this point, Carver or Bethany will ask to take the will back to Leandra immediately. Agreeing with the sibling will transport the party back to Gamlen's house.
Results
Leandra will be pleased to find out that her parents did not die angry, and will try to petition with the Viscount to get the estate back.
Rewards
Entering the vault in the Amell Estate basement you gain Bethany: friendship (+15).
A follow-up conversation with Bethany can be started after the quest and you can gain Bethany: friendship (+10) by choosing Diplomatic/Helpful dialogues or 
(+10) if choosing Aggressive/Direct dialogues.
Notes
- If you leave the cellar in the middle of the quest, the entrance to the cellar will not be displayed on the minimap. It can still be used to enter the cellar however.
- Above the locked door to the Amell vault is the skull of an Ogre. It is never specifically stated who was responsible for placing it there.