Raerek

Raerek is the Steward and uncle to the Jarl of Markarth. He previously served as Steward to the last Jarl. He will offer you the option to buy Vlindrel Hall and optional decorations once the Jarl approves.

Bugs
Sometimes even after being offered to buy a house, the option will disappear and you will be unable to buy a house

My solution.

I had this issue, I needed to blackmail him for the Stormcloak quest. Looks like a lot of people had the issue, or something similar with this NPC. Not many answers however, most people just post how to do the quest ( the quest is a little difficult to do, but I honestly thing it was the bug that was causing most of the troubles, its really not that hard to figure out what to do), or say they had no problems.

Anyway, I solved it by becoming a Lord/Thane in the city.

I had already visited the Stormcloak quest giver 3 times, and he kept telling me to go back and get evidence, wich I actually had on me the 2nd and 3rd time I spoke to him, but the quest wouldnt update and move the quest marker to Markath. After becoming a Jarl and speaking to him a 4th time, the quest marker finally updated and I was able to blackmail Raerek.

So my hint/suggestion to anybody stuck here is this : Keep returning to the wherever the quest marker tells you to go, and ask again what you need to do, and hope the quest triggers the next step. You are not able to talk to Raerek about the Talos amulet with out the quest update.

I was however able to buy my new home before the 4th visit to try the Stormclak quest-giver again.