If you want to buy a house in Skyrim, you need to become a Thane.
Whiterun's Thanehood comes with the first parts of the main quest and is relatively straight forward to get, but I've been having trouble attaining Thanehood in the other cities.
How do I become a Thane in Solitude, Windhelm, Riften, and Markarth? Are there specific quests that need to be done to earn it? (Like in Whiterun?)
Best Answer
In general, getting the title thane bestowed upon you follows a set strategy:
The Eastmarch (Windhelm)
Falkreath
Haafingar (Solitude)
Hjaalmarch (Morthal)
The Pale (Dawnstar)
The Reach (Markarth)
The Rift (Riften)
My complete walkthrough for this is contained in another answer about how to get the house in Riften, but since that's the final requirement for obtaining the title, it's exactly the same. In short:
Whiterun
You've already mentioned how to become thane here, but for the sake of completeness:
Winterhold