You'll receive this quest after your first encounter with Varric. Your task here is to talk to him inside The Hanged Man. You can find The Hanged Man in the middle part of Lowtown. 

You must first find your way into The Hanged Man in Lowtown. Inside ,initiate a conversation with  your  companion Varric. In this cut-scene Varric will express the need of a better entrance to the Deep Road. Your first set of dialogue options has no bearing so you may choose whichever you like. After that he'll mention his new information about a Grey Warden in the city. Here is the list of you dialogue options and their respective companions relationship results.

Investigate - Choose this to ask more questions about the information he received about the Grey Warden or other way into the Deep Roads.

After the sequence of conversation you'll complete this quest and update your quest The Deep Roads Expedition. You will also gain a codex entry The Grey Wardens.

MAIN PLOT UNLOCKED -  Tranquility

RUMORS UNLOCKED -  Wayward Son

RUMORS UNLOCKED -  Act of Mercy

RUMORS UNLOCKED -  Enemies Among Us

RUMORS UNLOCKED -  Shepherding Wolves

If this is your first time in The Hanged Man, You may explore the area for additional loots and codex entry.
(1) Crate, (2) History of Kirwall: Chapter 1 - codex entry, (3) Crate , (4) Barrel