When Alduin retreats after your battle, you need to find out where he has gone if you are to face him again and finish him off for good. One of his dragon minions would probably know; if you could manage to capture and interrogate one.
Talk to either Paarthurnax, Arngeir, or Esbern to get ideas on how you might capture a dragon and find out from it where Alduin is now. You can talk to all three, but only one is needed to continue the quest as they all come up with the same idea: Dragonsreach was once a prison to a dragon, who must have been captured alive somehow. The Jarl of Whiterun would surely know how that was accomplished. You should go and speak to him.
If you speak to Esbern, he will also tell you that he knows who the Greybeards’ leader is, and he wants you to kill Paarthurnax for the dragon’s past crimes. This will start the optional quest Paarthurnax.
When you go to Whiterun and talk with the Jarl, what happens next will depend on your progress in the quest-line of either faction of the civil war currently taking place in Skyrim:
- If the civil conflict has been settled in favor of either the Legion or the Stormcloaks, the Jarl will give you his full cooperation.
- If the civil conflict hasn’t been settled for either side and is still ongoing, the Jarl will reluctantly tell you that he can’t capture a dragon while the ongoing war is threatening his hold. If you want him to help you in this, you will have to get both sides to agree to a truce. He suggests that the Greybeards should host the necessary negotiations.
- Should you choose to complete the quest Paarthurnax before the negotiations, and anger or kill Arngeir, you will have no way of negotiating the truce and will have no choice but to end the civil war through battle in favor of one of the sides.
If events require you to negotiate a truce between the Legion and the Stormcloaks, the quest Season Unending begins immediately, and this quest will resume once you have completed that one.
Once the Jarl of Whiterun has agreed to help you capture one of Alduin’s minion dragons in Dragonsreach, you still need to find a way to lure a dragon to that ancient stronghold. You can think of two individuals who might have an idea how: Esbern and Paarthurnax.
Esbern has lately been busy in the Sky Haven Temple library. It appears the ancient Blades recorded many of the names of the dragons they slew. By cross-referencing the burial-site map that Delphine created from the Dragonstone, Esbern has identified one of Alduin’s raised dragons.
Paarthurnax, with his knowledge of the individual members of his own kind, and of the ways of the dragon shouts, tells you about a specific dragon you should capture and interrogate and can teach you a Shout you can use to lure him to you.
Because the names of dragons are always three Words of Power (Shouts), the dragon will hear and come to you when his name is bellowed. The dragon you’re going to try to capture is named Od-Ah-Viing (“Winged Snow Hunter”), and you learn to shout his name in the dragon tongue to bring Odahviing to you.
NOTE: Delphine, if she’s with Esbern, assures you that the Blades won’t be nearly as accommodating in the future if you don’t kill Paarthurnax as punishment for his past crimes. This is another hint to start the quest Paarthurnax, but it remains completely optional.
Now that you’ve learned the shout that will draw the dragon Odahviing, you can use it anywhere you like. If you do, Odahviing will appear in the skies, but he’ll be too far up to bring down with Dragonrend. There is a facility at Dragonsreach stronghold that can be used to capture this dragon, just as it did for another dragon once before.
Return to Whiterun and inform the Jarl you’re ready to proceed. Follow him up some steps to the side of his throne and out the doors to the Northeast. You emerge on the large stone battlements of Dragonsreach. When the Jarl tells you his men are ready, go and stand where he indicates and shout Odahviing’s name. In a few moments the dragon will arrive.
As soon as he gets near enough, use Dragonrend to force him to land on the battlements. He’ll begin to advance towards you while attacking. Keep backing away from him so that he enters the large hallway structure and passes between the two huge chains attached to the giant stocks contraption. When the dragon moves forward far enough, the stocks slam down, trapping Odahviing.
Speak with him and demand to know where Alduin has fled to. Odahviing reveals that he has traveled to Sovngarde, the Nord afterlife, to regain his strength by devouring the souls of the dead, and in time will return even stronger than before. You cannot allow that to happen. You must stop him before he makes himself stronger.
Odahviing says that a portal to Sovngarde is located at Skuldafn, one of Alduins ancient temples in the Eastern mountains. Unfortunately, the only way to gain access to this place is through the air. Odahviing offers to fly you there if you will release him.