New main objective: Talk to Paarthurnax or Talk to Arngeir or Talk to Esbern
You can talk about Alduin's probable hideout and ideas on finding it with one of three different characters. The choice doesn't really matter, though choosing Esbern will let you activate a certain side quest faster. Of course nothing stands in the way of visiting the two other characters anyway. The most obvious choice is talking with Paarthurnax (screen above), as you won't even have to leave The Throat of the World.
The second character that you can ask for help is Arngeir. The Greybeard can be found inside the monastery at the High Hrothgar (screen above).
The last suggested character is Esbern, who you can find at the Sky Haven Temple. Esbern won't probably be beside the Alduin's Wall, but beside the observation point outside (screen above). Regardless of your choice regarding who to talk with, you will learn more about Dragonsreach in Whiterun, originally built with catching a dragon in mind and that in order to find Alduin you will have to catch a different dragon and coax him into working with you.
If you decided to meet with Esbern, the conversation will have an additional segment. Esbern has learnt that Paarthurnax is a dragon and will inform you that in the past he was Alduin's right hand, responsible for unimaginable crimes. The old man will require you to kill him and your further cooperation with the Blades will depend on it. A new side quest, Paarthurnax will appear in your journal and as it's partially connected with the main plot, you can find its description in the further part of this guide. Don't worry if you talked about Alduin with Paarthurnax or Arngeir, as you will be able to unlock this quest later on. You can also gain it by talking with Esbern independently or by speaking with Delphine after finishing the negotiations as part of one of the further quests - Season Unending.
Side quest unlocked: Paarthurnax