You must assist five people in the hold, usually by completing favor quests for them. Once you have completed the prerequisite, speak with the jarl again and he will ask you to assist the people of the hold. If the jarl is Thongvor, whether he was installed via Season Unending or The Battle for Fort Sungard, then the completion of that quest is sufficient to fulfill the prerequisite. If the jarl is Igmund, then you must usually kill a Forsworn leader and Find Hrolfdir's Shield however, if he was ousted from Markarth during Season Unending and reinstated following The Battle for Fort Sungard, that is sufficient to begin this quest. In order to receive this quest you must first aid the current jarl. Once this is achieved you can become Thane. The jarl requires you buy the house in town as well as performing five favors for the townsfolk. This is an umbrella quest for several other quests that must be completed if you wish to become Thane of the Reach. The jarl can be found in multiple places - arguing with Calcelmo, roaming the halls, sleeping in his bedroom or sitting on the throne. The jarl lives in Understone Keep in Markarth. This is a radiant quest which you can receive from Igmund or Thongvor Silver-Blood, whichever is Jarl of the Reach.

Skyrim thane of the reach