Why won’t Farkas give me any work to do and is it related to this possible Purity quest bug

This a known bug caused by accepting Vilkas's Purity quest before Farkas. To resolve it accept and finish Farkas's Purity quest first and be sure that the quest giver is following you the whole way through.

To avoid bugs on this quest

• First speak to Vilkas. He gives you a random quest, then he gives you his Purity quest (Remember to have the heads in your inventory). Decline his Purity quest. If Farkas doesn't give you any quest after that, take an Aela's random quest and complete it. This will unblock Farkas random quest. Do it and then complete his Purity quest. When thats done, Vilkas will complete his Purity quest without a problem.

• After you complete Purity with either Farkas or Vilkas , if you immediately go talk to the other one, sometimes the quest will show up in your completed quest section, and you will not be able to activate the altar in Ysgrammor's tomb. This can be solved by loading a save state from before you accepted the first Purity mission, re-completing it, then doing another mission. Any mission should do, you can take an Animal Extermination mission from Aela that does the job nicely. After doing another mission, talking to the second of the two brothers starts the mission the way it's supposed to and you can complete it, making both Farkas and Vilkas very happy with you.

Hope that helps :)

Nope, there is currently no other way, as this one was already difficult enough to come up with. Even on pc, there is no way to use the console to work it out. It is probably what you already found, but I will add it anyway in case someone else is looking for it:

According to The Elder Scrolls wiki:

Solution #1 - For adventurers who have not cleared Driftshade Refuge or completed the quest to find the Helm of Winterhold.

You may get the bug if you have the quest to retrieve the Helm of Winterhold for the Jarl of Winterhold, under misc quests, but have not completed the quest to retrieve the Helm. Load up on invisibility potions and sneak through the Refuge (killing no one), retrieve the Helm, and return it to the Jarl. Then return to Whiterun and Vilkas should give you the next quest, Purity of Revenge.

NOTE: You don't need to sneak or use invisibility potions in Driftshade Refuge. I ran through full speed ignoring everything until I got the helm and ran out. Turned the quest into the Jarl and THEN went to kill the witches. This wont work if you have already killed the witches and got the message to return to Kodlak. As long as you do NOT talk to Vilkas and finish Blood's Honor, you can repair the glitch in this fashion. -Correction- this CAN be done if the witches have been killed already and you HAVE NOT returned to Whiterun. Sneak through to retrieve the helm and return the Helm of Winterhold to the Jarl of Winterhold in order to complete the quest. After the quest is complete make your return to Jorrvaskr, and Vilkas should give you the next quest, Purity of Revenge. (XBOX). Confirmed x2 on Xbox 360. **** This also worked on PS3, I was detected twice inside Driftshade Refuge but I hid long enough for the enemies to go back to patrol mode.

Actually, you might want to describe more accurately at which point of the Companions quest you are (for example, if you already returned to Jorvasskr or not)