Bleak Falls Barrow is the third quest in the main storyline in the game The Elder Scrolls V: Skyrim
This quest is unlocked after completing the quest Before the Storm
Follow the Jarl to Farengar's lab in Dragonsreach.
Farengar is working on some project related to dragons. He wants you to find a Dragonstone, an ancient stone tablet with a map to dragon burial sites, from the ruins in Bleak Falls Barrow. The Dragonstone holds the clues as to why dragons are returning to Tamriel.
At this point, you can follow the walkthrough as listed in The Golden Claw Quest
It is possible you already have the Dragonstone if you had already done that quest, so you can immediately give it to him to complete the quest. If you do not have the Dragonstone, travel to Bleak Falls Barrow to recover it.
Go through the bridge and follow the marker northwest along the path to Bleak Falls Barrow
Enter Bleak Falls Temple
Head down some stairs past the main chamber, and you will arrive at a pillar puzzle
Each pillar has a possible shape: Snake, Bird or Whale
You can see the symbols on top and on a broker pillar on the floor. The correct combination is SNAKE SNAKE WHALE:
Cut him down, and he starts running away. Run after him. DO NOT LOSE SIGHT OF HIM, AS HE HAS THE GOLDEN CLAW WHICH YOU NEED
Arvel will either be killed by you or by the skeletons. Loot the golden claw from him. Be sure to also loot his Journal. This gives you the hints needed to solve the puzzle, if you didn't have a guide.
You now have to discover the secret of Bleak Falls Barrow. Head through to Bleak Falls Sanctum.
After passing through some pendulum traps, you get to a locked gate. Before that you have more Draugr to deal with. Make your way up to the second level and find the Iron Door. This leads to another Puzzle.
The combination of the gate is on the Golden Claw itself, you can see if you inspect it and zoom on it in your inventory.
From top to bottom, the combination is: BEAR MOTH OWL
Climb the stairs to enter a big room
Now make your way back to Whiterun. Just head outside by using the stairs to the left of the Word Wall. Pull the lever against the wall and it will reveal a hidden door. Head on through it and it is a short walk outside. Just Fast travel there after that.
This quest is followed by Dragon Rising