“Proving Honor” is a Companion Faction Quest in The Elder Scrolls V: Skyrim
Head inside Jorrvaskr and talk with whoever sent you on your last quest. They will tell you to find and speak with Skjor. Find him inside the Companion Hall. He will tell you that your time to prove yourself has come. He tells of someone finding a fragment of the legendary battleaxe, Wuuthrad. He wants you and Farkas to head to the location and reclaim the fragment.
Inside Dustman's Cairn, there are a number of inanimate Draugr scattered throughout the room. Also, on the table in front of you, is the skill book The Battle of Sancre Tor. You can also find a Novice-level chest to the left-hand side of the table. After that, start going deeper inside.
The second room has an Arcane Enchanter and 2 thrones. There is only one path available and it is blocked by a gate. Find the Northernmost room. Head inside there and pull the lever. This will spring a trap on you. Farkas will start looking for a way to free you but be confronted by members of the Silver Hand. He will transform into a Werewolf and dispatch them. Talk with him after that to learn a bit more about the Companions and the circle. AFter that conversation, that off to the South.
Before long, you will some stairs down. Take them, turn to the right. Look for the nearby table. There are some helpful potions to grab (and some gold). After that, press forward.
This next portion has a lot of sneaky Draugr. Look for any of them that are clothed in some fashion. Sneak up on them and attack to get some extra damage. From ther, just folliow the slow downslope as you make your way deeper. This will eventually lead you to the next part of the cairn: Dustman's Crypt.
Once inside the Crypt you have more Silver Hand and Draugr to deal with. head across the bridge. If you are notice, do not worry too much. There are a numebr of Draugr the Silver Hand have to deal with to reach you. On that note, if you are somehow managing to sneak though, look for any of the clothed/armored draugr to get them with sneak attacks. Follow the path as it winds its way down. This will eventually get you to the area you passed over on the way in.
When you make it down to the lower level, look for the chest. Search as you deal with the last few Sliver Hands in this area. You will find a locked Iron Door. If you search nearby and find the unlocked chest, you will find the Dustman's Cairn Key. Use it to open the door.
This leads to a straight forward hallway. Throughout it you will encounter Skeevers that you will need to deal with. When you make it to the room with the Alchemy Lab, expect a pair of Skeevers. Head to the Southern part of the room. You will find a locked door (Novice level). Pick it to find a heal potion. Turn to the East after that. Continue over to the East after that and follow the corridor for a few turns until you make it to the next room. Be on guard for more skeevers.
After you deal with the Spiders you want to head over to the East. From there, follow the upslope. This will lead to a pair of bridges that lead to a pair of Draugr. Defeat them and keep following the path in front of you. it will take you to the West and North, leading to a door. Open it up and head through. This leads to the final room of this dungeon.
Go to the far end of the room. You will find a Word Wall waiting for you there. Claim a word for the Fire Breath Dragon Shout. Just in front of that is the table with a Fragment of Wuuthrad. Once you grab the fragment, you will need to fend off close to a dozen different Draugr. This does include at least one or two Draugr Wights nad Restless Draugr.
After you make it through all the undead, a passage will open on one of the sides. Follow the corridor to make your way back to Skyrim itself. From there, just get back to Whiterun.
When you make it to Whiterun, head for Jorrvaskr. Do not go inside. Instead, go around to the training yard. You will find the whole group has assembled. Approach them to begin your initiation into the Companions. When it is does, talk with Eorlund Gray-Mane at the Skyforge to get a piece of Sky Steel of your own.