I talked to the Jarl but he's telling me that I need to prove myself. I don't see an active questline related to this at all. Did I miss something?

http://www.uesp.net/wiki/Skyrim:Vlindrel_Hall

Talk to his steward (guy sitting next to him).

You have to be at least level 20, do two quests for the jarl, then you can buy it.

I dont remember doing any quest. I went there to buy a house after I finish the civil war quest.

Believe it is a clear out forsworn or something similar kind of quest

Vlindrel Hall is a great home. Beautiful Dwemer architecture and a lovely view. Markarth is an amazing city!

Eilkno: Now that the post has been moved I can say: My first go-round I gave Markarth to the stormcloaks via the peace treaty and was able to purchase Vlindrel Hall straight away from the new jarl.

Definitely, one of my favourite houses as well! I love the architecture of the city and how it is laid out. Correct me if I am wrong but there has never been a city quite like it before.

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I will move this now.

Yeah, it is a really cool looking city - I'm not a fan of the corruption, but I'll svck it up. I find it interesting that it's the only above-ground Dwemer city or is it just the only surviving above-ground Dwemer city?

Haha I'm not too sure to be honest . Although I do like how it is so vertical yet easy to navigate, and the water running through and the waterfalls make it an amazing city!

Yes, it's the way we romanticize (and possibly how they actually were) ancient Greek cities/city-states.

Yea my favorite city, with my favorite quest. Ever sense i saw that city before skyrim came out i said " One day, im gonna live there".