Whichever potential leader you chose to support will send you into the Deep Roads beneath Orzammar to find Branka and ask her to support your choice for king. After fighting legions of Darkspawn and other creatures, you'll find Branka. She'll get you to complete a series of trapped rooms to get to the Anvil. When you complete the rooms, you'll find Cairidin. He'll ask you to destroy the Anvil. Whether you choose to destroy it or not, you will have the means to put into place the new king.
By completing this and taking this path, you'll open up "A Golem's Memories" if you have "The Stone Prisoner" DLC.