After completing the quest "Take Up Arms" and doing some additional work for any of the Companions,
Locate Skjor in Jorrvaskr. Tell him you are looking for work. He will tell you they have something special for you. Ask him what he means. He will say last week a scholar came to them, and he said he know where they could find a fragment of Wuuthrad. Tell him you will be honored to retrieve it.

This will be your trial. Do well, and you will be counted among the companions. Farkas will be your Shield-Sibling for this mission.

Open your quest journal.
Proving Honor
The Companions have given me one final challenge before inducting me. I'm to retrieve a fragment of Wuuthrad, the Blade of Ysgramor, from Dustman's Cairn while Farkas judges my valor.

Speak to Farkas.

Locate Farkas. Talk to him to get some info on the trial. Ask him what is Wuuthrad. He will tell you that Ysgramor was the hero who started the Companions, and Wuuthrad was his weapon.

Tell Farkas  you will meet him at Dustman's Cairn. Your objective now is to retrieve the fragment. Exit Whiterun and travel Northwest to Dustman's Cairn. Jump down the pit and enter through the door. Farkas will be waiting for you inside.

Go East down the crypts. Farkas will  observe that it looks like someone has been digging here recently, so better watch out of enemies. Arrive at a room with a desk that has a Two-Handed skill book on it, as well as a novice chest which you should be able to open easily. Keep heading east into the tunnel, then south, and east again. You will run onto some Draugr, defeat them.

Turn north, open the wooden door. Head east into a big circular cavenr with a throne. To the SE is a locked portcullis. Enter the small room to the NE and switch the lever to open the gate. This will lock you inside the small room! Farkas will tell you not to worry, he will find the release. Suddenly members of the Silver Hand appear and attack Farkas! He seems outnumbered and surrendered, so there is little hope. But then he transforms into a werewolf and kills them all!

He opens the gate and comes talk to you. He tells you that he hopes he didn't scare you. Ask him what was that. He will say it's a blessing given to them. They can be like wild beasts, fearsome. Ask him if he is going to make you a werewolf. He will say only the Circle has beast blood. He says you have got to keep moving.

The tunnel to the SE should be unblocked now, take it. To the south, there will be a pedestal with a helmet on it. Take the helmet but watch out for the trap. There will also be a flawless amethyst here if you need it for Talen-Jei. Some more Silver Hands will appear, kill them.

At the end of the tunnel there will be a door to the north, go through, and take another door to the east. Enter a chamber with some Silver Hands in it. After you defeat them, there is also a novice chest under the stairs. Climb the stairs. To the east there is a Master chest.  Head south and down the stairs. Open the wooden door. Make sure to loot Burial Urns around as they have gold in them.

Go east, then south, then east. Neutralize any Silver Hands or Draugr on the way. Reach a room with an Adept chest. Go down south, and through the wooden door. Defeat the silver hand in the grotto ahead, and take the door to the south. Clear the Silver Hands in the next chamber and enter Dustman's Crypt. Open the door through the west and pass through a gated tunnel. Work your way down the corridors, then go east and enter another big chamber. Pick the lock of the chest under the stairs if you wish, then you can also pick the door to the south which hides some treasure.

Enter a throne room to the North, and find the Dustman's Cairn key in a chest. Use it to unlock the iron door to the north. Reach a small chamber and go through the door to the east. Work your way across the narrow corridors. Reach a cave with a Frostbite Spider in it. Go east and climb up the ramp to the north. Defeat the restless Draugr along the way.

Go west, then south. To the west you will now see an iron door, open it. Enter  a huge throne room. Go to the glyph wall to the north, and absorb the words, Word of Power: Fire, Fire Breath. Take the Fragment of Wuuthrad from the table. Your goal now is to return to Jorravskr. Loot the rest of the treasure in the room

Climb up the platform to the south. Suddenly all the Draugr start coming out of their tombs! There will be a lot of them. Kill them all, then exit the chamber to the south, Climb out the tunnels, activate the lever, and work your way back out to Skyrim. Fast Travel to Whiterun and enter Jorravskr.

Vilkas will be waiting for  you near the entrance. He says he has been awaiting your return. Ask him why he was waiting for you. He will tell you to follow him.. He will lead you into the courtyard where the rest of the Companions await. They will welcome you to the Circle.

"I successfully retrieved a fragment of Wuurthrad, a blade of Ysgramor, to earn my place in the Companions".

This complete the quest "Proving Honor". Talk to Skjor, Farkas, Aela or Vilkas to do another faction quest before embarking on the next quest, "The Silver Hand".