As you enter the Riverwood Trader, Camilla and the shopkeeper are having an argument. When you talk to the shopkeeper, Lucan Valerius, he tells you that he is having some problems with bandits. There was a break in. The robbers were interested in only one thing. An ornament, solid gold, in the shape of a dragon claw. Tell him you will help him get the claw back. He has some coin coming in and offers it to you as a reward if you recover the claw.

He tells you that to find those bandits, you want to head to bleak falls barrow,  North west of town. Camilla offers herself as a guide, although Lucan insists she accompanies you only to the edge of town.

Open your quest journal to see info on this quest:

The Golden Claw
Lucan Valerius in Riverwood is offering me a gold reward if I retrieve his antique golden claw, stolen by bandits camped in Bleak Falls Barrow.

Retrieve the Golden Claw.

Camilla leaves the shop, you should follow her, although first you should make room in your inventory and get some gold by selling some items to the shopkeeper. Ask him what he has to sale to open the Barter menu. Items above the divider line belong to the vendor. Items below the line belong to you. To buy or sell an item, select it and press E.

After  you are done Bartering, leave the shop and find Camilla waiting for you outside. She will tell you that you have to go through town and across the bridge to get to Bleak Barrow. The thieves must be mad for hiding out there, as the crypts there are filled with rats, trolls, and who knows what else. She will also mention that it's strange that the robbers stole only that artifact, when there are lots of things in the store worth just as much. Lucan found the golden claw a year ago when he opened the shop, and he never told Camilla exactly how he got it. Follow her. She will lead you to the bridge on the edge of town.  The path to the Northwest up the mountains will lead you to Bleak Falls Barrow. Camilla will now turn back and return to her brother, Lucan.

Cross the bridge, then take a left turn and follow the road up the mountain to the NW. You can check your map every now and them to get bearings on your position.

On the way, you will pass through a tower guarded by bandits. Defeat the bandits and climb up the tower to find a chest full of gold.

Keep going towards your waypoint, and you will reach the crypts of Bleak Falls Barrow. Climb up some stairs and defeat the bandits guarding the entrance. Go to the door and enter the Bleak Falls Temple. Explore the entry hall, kill the enemies there and picklock the chest for some gold. head down the stairs past some cobwebs. Loot the Burial urns for a little more gold.

Continue working your way to the bottom of the crypt. You will get to a chamber with some pillars in it. This is a PILLAR PUZZLE

There are three pillars, and each pillar has three sides with a different animal symbol on each side: Snake, Bird or Fish.

To solve this, look around the room. Above the gate, you will see three symbols: A Snake and A fish, and in between them there will be some ruins instead of a symbol. If you look on the bottom floor, you will see the missing symbol, a snake. So set the pillars in that same order - from left to right - SNAKE, SNAKE, FISH. Then pull the level and the gate will open. If you got the symbols wrong, you will set up a dart trap. (SEE PICTURE BELOW FOR SOLUTION)

Skyrim Pillars Puzzle Solution

Enter a room that's filled with some treasures, including a "Thief" book that will increase your pickpocket skill. Go down the spiral stairway. You will be attacked by some rats in the bottom. Keep going, and you will reach some impassable cobwebs. Destroy them with your weapon.

There is a big wounded frostbite spider inside so be prepared. This battle can be tough. kill it. You will hear some cries for help from Arvel the Swift. He is trapped inside some spider web. Go talk to him. He will ask you to cut him free. Ask him where the golden claw is. He will tell you about how the claw can  unlock the hall of stories and how he will show you its secret once you release him. If you ask him to hand over the claw first, he says he cannot move and to cut him out of there.  Cut Arvel down.

But as soon as you do that, he starts running away! He is "Arvel the Swift" after all. Chase him and get the claw for him whatever it takes (in other words: kill him). Now some skeletons appear! kill them and loot for some gold.

Your new objective, now that you have the claw, is to find the secret of Bleak Falls Barrow. Head all the way west and then north. Watch for the trap on the floor that will cause a door to close on you and injure you. keep going north, then turn right. Bring up the map to see your destination.

Enter a room with some tombs in it. Skeletons will rise and attack you! kill the skeletons.

Next, cross the corridor with the two swinging pendulums in it. Time your approach carefully, as to not get hit by the pendulums. At the right moment, move through one pendulum and stop. Then clear the other two. Make sure to save before going through this obstacle.

Head down some stairs. Keep going through the corridors, and be prepared to fight a lot of skeletons on the way.

When you reach a room with a waterfall, activate the chain to the south to lift the gate. Head south along the water, then East towards the area with the light. Climb down into the room with the vapor, then climb up onto the exit to the North  through the tunnel. Check your map to get some bearings.

Go west, then North again and you will find a room with some light, as well as an enemy skeleton. Kill it and open the door to the Bleak Falls Sanctum. Enter the sanctum and work your way through the tunnels, past another pendulum trap.

Head up some stairs and past a bridge, and arrive at an iron door. Open it, and cross the hallway to find a door with the dragon claw symbol! This is another PUZZLE

There is an inner ring, a middle ring, and a outer ring. each of them has a butterfly, bear, and owl symbol. Solution: Bring up your inventory and examine the gold claw. Look carefully. It has the order of the symbols on it! Put the owl in the inner ring, then the butterfly in the middle ring, then the bear in the outer ring. Activate the Keyhole using the gold dragon claw (If you are not carrying the claw, you missed a step. Read a few paragraphs back on how to get it). The door will open. (SEE PICTURE BELOW FOR SOLUTION)

golden claw door puzzle solutions skyrim

Note: If the glyphs are not rotating, reset the puzzle by going back through the bridge and out the door, then come right back in. That will reset the puzzle and rings should turn again.

Climb the stairs and you will enter a big chamber. Head to the glowing blue glyphs. They will say "word of power, force unrelenting force". Reading the glyphs will uncover the secret of Bleak Falls Barrow and complete the objective, and will teach you a new spell, Unrelenting Force. This shout is powerful but requires dragon souls to unlock.

Turn to the chests behind to loot them, but you'll have to kill the Draugr Overlord that pops out of one of them first.

IMPORTANT: Make sure to kill and LOOT the Draugr Overlord. He carries a DRAGONSTONE, an ancient stone tablet with a map of dragon burial sites. If you are currently on the Bleak Falls Barrow quest from Farengar in Whiterun, this is the artifact you need to complete that quest. If you have not gotten that quest yet, you will be able to complete the quest immediately upon receiving it from Jarl and Farengar just by handing over this Dragonstone to them. Do NOT leave the sanctum without the Dragonstone!

After you have killed the overlord, loot the chest (there is some good items in there!) and exit the sanctum through the west, activating a lever to open the door. Once you are out, bring up the map, locate Riverwood and select it to fast travel to it.

Go to the Riverwood trader and give the claw to Lucan. This will complete the quest and Lucan will give you a nice reward of gold coins.