On your initial arrival at Markarth, you won't get your usual chance to unwind and do whatever before something important comes up and gets slapped in your face- instead, you are immediately faced with an event that starts off a quest line.
As you enter, a woman by the name of Margret is standing in front of the jewellers. You will get a front row seat to watch as a man steps up and tries to murder her, shouting about gaining justice for the Forsworn. You basically have two options here- watch Margret die or save her by killing her attacker first. If you watch her die, you can still help the guards kill her attacker, but rather than looting the apparent nutjob and recieving a silver necklace in thanks from the woman, you will find yourself with two corpses to strip.
Either way, the Markarth guard arrive almost immediately, and once the whole situation is resolved, they will start telling you that there's nothing else to see here and you should go about your business. Oddly, when one of them talks to you face to face, he will outright deny that the Forsworn had anything to do with this or even could have, and rather strongly 'recommend' that you leave off of this supposedly isolated incident.
All of the others present (including Margret) give similar impressions of being upset, distressed, and generally confused about events. The only notable exception to this is Eltrys, a Nord man who approaches you after the guard has moved on to let you know that you dropped 'this paper', which he hands to you. No matter how you speak to him, he insists that the paper is yours (possibly with a vague warning about being too obvious about things) and then walks off.
On opening the note, you will find that it instructs you to go to the Shrine of Talos in Markarth, whose door can be found in the passageway beneath the Temple of Dibella located in the central rock spire of the city. At this point, you have an interesting choice to make- Eltrys will give you several tasks to do once you talk to him, but you don’t actually have to go see him in order to do these things. He will give you a modest reward each time- 200 gold is the base amount per task, though it scales with your level and a level 30-ish character will often pick up a thousand gold or more per piece of information- but you may consider that an insignificant amount of gold, or simply not worth the time spent backtracking to him each time you do something.