Kodlak Whitemane wants the Dragonborn to help him cure himself of his Lycanthropy by acquiring the heads of the witches that tricked The Companions.
Speak to Kodlak Whitemane
Aela the Huntress tells the Dragonborn that Kodlak Whitemane has found out about their conflicts with the Silver Hand. She says that Kodlak wants to speak with the Dragonborn.
Collect a Glenmoril Witch's Head
After talking to Kodlak, a map marker will be given for the location of the witches. The witches reside in a cave called Glenmoril Coven near Bilegulch Mine in the southwest corner of Skyrim. Upon entering the cave there will be several witches sporadically placed in the mine. The first one will be very close to the entrance.
(Optional) Wipe out the Glenmoril Witches
While the quest only requires the death of one of the Witches, it may be of benefit to kill all five. Later on after curing Kodlak of Lycanthropy, the Dragonborn may cure themself, provided a second head has been acquired. Afterwards, the quest Purity requires two more heads, as Vilkas and Farkas wish to be cured as well. The fifth head is unused as Aela will choose to remain a werewolf.
Upon return to Whiterun, several citizens can be noticed gathered around near the stairs leading to Jorrvaskr. Halfway up the stairs, Torvar and Aela the Huntress are found next to dead Silver Hand members. After entering the building, Vilkas seeks the Dragonborn out and asks where was the Dragonborn during the time of need. In the Dragonborn's absence, the Silver Hand attacked Jorrvaskr, managed to kill Kodlak and take off with all the fragments of Wuuthrad. Kodlak’s dead body is near the central fireplace, with Farkas and Njada Stonearm next to him. Vilkas claims that his death will be avenged, which ends the quest and immediately starts the next one, Purity of Revenge.
Kodlak Whitemane has caught wind of my secret missions with Aela and has asked to speak with me.
Objective: Speak to Kodlak Whitemane
As part of his quest to cure lycanthropy, Kodlak Whitemane has sent me to kill the Glenmoril Witches and bring back their heads.
Objective: Collect a Glenmoril Witch's head
Objective: (Optional) Wipe out the Glenmoril Witches
Objective: Return to Kodlak
I killed the Glenmoril Witches at the behest of Kodlak Whitemane, but he was killed by the Silver Hand before I could return.
Click to see the list of bugs encountered
360 At the start of this quest you must first speak with Kodlak Whiteman to obtain the quest. When you initiate dialogue though it will glitch and not enter the quest discussion.
When you enter Jorrvaskr, Vilkas will not talk to you, which makes it impossible to end the current quest, and to start the next one.
If Njada Stonearm is killed before this quest, it becomes uncompletable due to her background dialogue. (02/08/2012: As of the 184.108.40.206 patch on the 360, if Njada Stonearm is dead at this point, it does not break the quest and this part of the bug seems to no longer apply.
When returning after retrieving the heads, both Vilkas and Farkas keep initiating dialog with you through the final few quests. This gets quite annoying as you keep turning around to speak to one or both of them. There is currently no clear way to get around it while continuing the quest. If you fast travel away, it will fix this bug temporarily, but they will resume the dialogue back a Jorrvaskr.
Forever Mourning Bug
This is a bug that can prevent further Companions quest advancement. When returning from killing the witches, the adventurer will enter Jorrvaskr and find the aftermath of a battle where Kodlak has died. Vilkas will seek you out as normal and complete this quest. However, he will then enter into a relaxed state without giving you the next quest, Purity of Revenge. Several other Companions will enter a permanent mourning state. In addition, some extra jobs obtained from the Thieves Guild will not be doable, as some of the houses get barred up with wooden stakes during the attack that you never see. For example, a "Burglary Job" obtained from Vex to grab three things from someone's house will be incompletable, but you can always quit a job you don't want to do. It will also fail to show up on the map when selected.
The state of permanent mourning can be broken if you attack a companion they should all get up and attack you. At this point you should sheath your weapon to yield. This should enable interaction with the companions again. If they do not accept your yield then you should run away and fast travel far away as soon as you can. Once you do this you should wait for a day or two before returning. You should be able to speak to the companions again and get missions from them. Unfortunately, despite enabling interaction, this still does not seem to allow you to begin purity of revenge..
Solution: For adventurers who have not cleared Driftshade Refuge or completed the quest to find the Helm of Winterhold
PC If the witches have been killed and returned to Whiterun but have not retrieved the Helm: Go to the Refuge and run through it without attacking. Enemies will engage, but with enough health, stamina, and armor you should be able to run through without dying. It may take a few tries to learn the layout well enough. Retrieve the Helm and run out, then travel to Winterhold and return the Helm. Travel to Whiterun, find Vilkas and initiate a conversation. Open the console (~) and type "setstage c05 10", press enter and close the console. Vilkas should join you and Purity of Revenge will automatically start. This method worked on PC version 1.3.10.
PC Solution: If the witches have been killed and returned to Whiterun but have not retrieved the Helm (If you have regained Winterhold in the Reunification of Skyrim): Since Jarl Korir has been driven out and replaced, he will not accept the helm and will instead talk about how he was driven from his home. Travel to Whiterun, find Vilkas and use the console command "setstage c05 1", press enter and close the console. Vilkas will join you and Purity of Revenge will automatically start.
Solution: For adventurers who have cleared Driftshade Refuge and completed the quest to find the Helm of Winterhold. Wait for 32 days before killing ANY witches for "Blood's Honor". Driftshade Refuge will still read as "cleared" on the map, but the enemies inside will have respawned at this point, allowing the quest "Purity of Revenge" to start as normal when you complete "Blood's Honor".
PC Solution: Console Command fix. To reset Driftshade Refuge, enter the console and type "resetinterior driftshadesanctuary01" and then "resetinterior driftshadesanctuary02" to reset the dungeon. Note, this only affects the dungeon once you enter drift shade refuge, not once it is typed, so to complete the reset you must enter the dungeon. Once you enter the dungeon, type "setstage c05 00" to start the quest for revenge. Alternatively, reset the dungeon, travel there to cause the reset to load, and then complete Blood's Honor allowing the quest to begin normally. Fast travel to Driftshade Refuge, enter the building and find the Silver Hand leader. He is the only enemy in full plate armor. Open the console, click the deceased enemy and type "resurrect". This will bring the enemy back to life. Do not attack him, instead leave the building. Once outside, open the console and type 'player.setstage c05 0'. Ref ID of Silver Hand Leader in my game was 0006a99c. You may be able to resurrect him without looking through the entire dungeon for his body. Try typing 'prid 0006a99c' followed by 'resurrect' in console, please confirm if this worked for you.
Solution: If you happened to just stumble upon Driftshade and clear it out, you will have to wait the 32 game days for the NPC to respawn.