Favor and Fault is a companion quest for Aveline in which the old Captain of the Guard, Jeven will be taken care of once and for all.
Acquisition
You will receive this quest from a letter in your estate mansion. The letter is sent by Knight-Captain Cullen.
Walkthrough
Speak to Aveline in her office in the Viscount's Keep. A conversation between Aveline and a templar is taking place. You will speak to Aveline about complaints made against her. It is said that she is coddling certain guards. You can either call the complaints true or not, that's your personal choice. Aveline suggests that you accompany her during Donnic's patrol at night in the Docks. Make sure Aveline is in your party to start this quest at the Docks.
When you arrive at the Docks, follow the marker on your map that points to the East. Guardsman Donnic and several guards, will be attacked by several Coterie thugs and alchemists, step in and help Donnic. After the fight a cutscene will follow. You can ask if Aveline is coddling the guards or you can comment on the fight Donnic just had. After the conversation, you will need to speak with Cullen in the Gallows Courtyard.
Brennan
Cullen will tell you that the complaints were made by anonymous people from Lowtown. Aveline wants to talk with Guardsman Brennan, whom she suspects is behind all this. Travel to Lowtown at night and again make sure Aveline is in your party. After talking to Guardsman Brennan, you will hear that the former Guard-Captain Jeven is behind all this. Travel with Aveline and your party to the rally held in Darktown.
Jeven
Travel through Darktown and kill all the thugs and mercenaries you encounter on your way. You will find Jeven on the south side of the map. You will need to kill Jeven, the insurgents and the mercenaries after the conversation. Loot Cornerstone and Aveline's last armor upgrade, Deflecting Joints - Guardsman Pattern from Jeven, and Hands of Glory from a nearby chest. Return to the barracks in the Viscount's Keep to talk with Aveline about Jeven's death.
Aveline: friendship (+5) after the conversation in her office, if you disagree with the complaints made against her. You also need to say you already know she's not doing anything unlawful. Or 
Aveline: rivalry (+5) if you ask her if the complaints are true.
Aveline: friendship (+15) automatically after the fight with Jeven.
Note that, when talking to Jeven, Hawke must select the option that allows Aveline to speak for herself. If Hawke makes any statement, Aveline: rivalry (+10) and Aveline's ingratitude, is gained instead.
If you had chosen to kill the Qunari yourself during Offered and Lost (aggressive response), Hawke will have the option of recognizing that some people in the current crowd were also present in Ser Varnel's Refuge... The resulting reaction from Jeven is fairly amusing.
Note: While fighting with Jevan you can run back to where Anders is and he will join you in the fight
Bugs
If you finish the main quests up to the confrontation in the Gallows before completing this quest, you will be unable to advance to the conversation with Cullen. You can use Aveline's plot debugger to skip ahead in the developer console.
Encountered a bug where if Jeven is killed at Darktown (Night) it doesn't register correctly. To remedy this, just leave and kill him in Darktown (Day).
Aveline's last armor upgrade, Deflecting Joints - Guardsman Pattern, should be found on Jeven. Save your game before encountering Jeven, Aveline's armor upgrade doesn't always drop.
Upon completing the quest, during the talk with Aveline, choosing the "He wanted a fight. He got it."-line caused Aveline's reply "The guards need to see me lead. I must have their confidence." to be spoken with a (female) sarcastic Hawke's voice, by Aveline. (Full friendship and I chose the sarcastic line against Jeven, rather then letting Aveline speak).