You must release Odahviing the dragon so he can fly you to Skuldafn temple, where there lies a portal to Sovngarde, the Nord afterlife, where Alduin has retreated following his battle with you to devour the souls of the dead in order to make himself stronger than before, strong enough to defeat you. You must reach him before he becomes too powerful.
In order to free Odahviing, talk to the Whiterun Guard standing on the second floor. Then go and speak to the dragon and he will fly you to Skuldafn temple on his back.
Upon arriving at the temple courtyard, you’ll encounter two dragons, and multiple Draugr Wights and Deathlords. Kill them all, or sneak past them if your Sneak skill is high enough, and enter the temple complex.
The first puzzle you encounter in Skuldafn will require you to open two portcullises. There is a lever nearby that will raise the portcullises, but it is currently not attached to the mechanism. In the upper chamber, there are three pillars with carved animal petroglyphs on them. Examine the two outer pillars first. On the outside of the ceremonial arch structure they are sitting under is a second carving, which is resting in the mouth of a stone head. Move the pillar on the western side so the Whale glyph is facing the chamber’s western side (in the same direction as the carving above the pillar). Move the pillar on the eastern side so the Snake glyph is facing the chamber’s eastern side. The mechanism is now attached.
Before you move the middle pillar, examine the two portcullises. Above each there sits a stone head, and in the mouth of each head is another glyph, partially obscured by the stone crossbeam. The left (northwest) mouth has a Snake. The right (northeast) has a Hawk. Now rotate the middle pillar so that the Snake or the Hawk faces the lever pedestal. Then pull the lever, and the portcullis corresponding to the Snake or Hawk glyph opens. You can have only one of these portcullises open at once. The passageway under the right portcullis leads to a blocked area with a small chest and the one under the left portcullis leads further into the temple.
You’ll soon come to an iron double door, on the other side of which is a second puzzle chamber. Clear the room of enemies and search the treasure chests before you focus your attention on the animal petroglyphs. Beginning on the ground floor, head east, and walk around the room until you spot a carved head with a Snake glyph inside its mouth. Return to the pillar facing the double doors you entered, and turn it so the Snake faces the doors (west). Ascend to the bridge area above. Locate a second carved head with a Hawk in its mouth. Match that so the Hawk is facing the same direction (outward or to the north) in the alcove underneath this head.
Turn around (to the north), and match the Whale on the pedestal in the opposite alcove with the final head above it. Then pull the lever on the pedestal. This will lower the bridge, allowing you to proceed.
Nordic Puzzle Door
You will shortly come across a Nordic Puzzle Door being guarded by a powerful Draugr. After you kill him, you can take from this corpse the Diamond Claw, an artifact which is the key to exiting this place.
Approach the Nordic Puzzle Door, but first inspect the palm of the Diamond Claw you just took of the corpse of the Draugr. There are three symbols etched into the palm of the claw: a Wolf, a Moth, and a Dragon. These correspond to the symbols on the door’s outer, middle, and inner rings respectively. Align the symbols on the rings of the door to match the appropriate symbols on the claw. This will cause the door to slide down so that you can proceed beyond.
Make sure you learn the Word of Power on the Word Wall in this next area before heading up onto the roof of the temple, because soon you’ll reach the portal to enter the realm of Sovngarde, after which point you won’t ever be able to explore Skuldafn again.
Once you step out onto the roof of the temple, head towards the jet of molten fire roaring into the skies. There is a dragon priest named Nahkriin who is in the process of speaking dragon tongue incantations to close the portal to Sovngarde. There are also two dragons on either side of you.
Engage Nahkriin and the dragons in battle. Slay the two dragons before turning your attention to the priest. Remember to use Dragonrend to bring them down to the ground where they are vulnerable to your attacks. Beware the electrical prowess of the dragon priest; use the arches and stairs as cover from his spell attacks if you need to. After he falls, if he managed to close the portal to Sovngarde before he died, don’t worry, the staff he was carrying will open it again. Also be sure to pick up Nahkriin’s Mask from his corpse as that is useful for the side quest Masks of The Dragon Priests.
Ascend the ceremonial steps. Activate the Dragon Seal at the top of the steps, and jam the shaft of Nahkriin’s staff into the seal’s center. A great plume of fire erupts from a whirling vortex – a gateway to the Aetherius realm, where Alduin cheats death and feeds off the souls of past heroes. Step down into the light to enter Sovngarde…