New Main Objective: Find evidence
Note! You get this mission during the Liberation of Skyrim main Stormcloak Rebellion Quest and its completion is required to advance to the next part.
Open the world map. Your destination is Markarth located in the west part of Skyrim (the above screen). If you've been there before, use Fast Travel. Otherwise you'll be forced to plan your journey; consider taking a wagon (for example the one from the Whiterun Stables).
When you reach Markarth, go west. You need to get to a monumental entrance to Understone Keep (the above screen). If you've been to the Jarl's keep before, you can travel there right away, especially since it's visible on the map.
Head straight ahead, going west and ignoring a guard you'll be passing by. Your destination is the Jarl's main residence. Ignore the people around because you don't need to talk to anyone at the moment. Instead, turn right, heading north. There is a passage leading to a part of the dwarven keep that is closed to guests (the above screen).
The amulet you're looking for is located in one the bedroom on the ground floor, but unfortunalety reaching it isn't easy. Notice that the nearby corridor is guarded by a sentry. You can ignore his few initial warnings, but after a couple of times he would eventually attack you. To avoid confrontation, I suggest sneaking or becoming invisible. In the latter case, you can use a proper spell or visit the closest store, The Hag's Cure, and buy an Invisibility Potion. No matter which option you choose, you'll reach a closed bedroom door (the above screen). If your thieving skills are high enough, you should be able to open them easily. Otherwise you need to think of something else.
My recommendation is to pickpocket one of the more important persons in the keep. If you choose the conditions of the theft well, you'll succeed even if you character doesn't specialize in this kind of things. The desired Key to Markarth Keep lies in the hands of, for example, Faleen or Raerek. As for the right moment, I suggest choosing either the morning of the evening hours. This is when the persons mentioned will be getting up (the above screen) or going to bed.
If you're having problems with stealing from a moving target, you can wait for another right moment, for example when one of them sits at a table to eat (the above screen). The most important thing is to crouch and be hidden. To make sure you succeed, save the game, especially if you haven't been raising the thieving skills before.
No matter if you have used the lockpick or stolen the key, you need to get to one specified bedroom. Close the door behind you and start examining it. Check the dresser by one of the beds (the above screen) and take Raerek's Inscribed Amulet of Talos. Return to main hall of Understone Keep quickly so as not to encourage the nearby guard to engage you in a fight.
Note - if you spend too much time exploring the areas closed to guests or run around the keep with your weapons drawn, you can encourage the guards to attack your character. If so, I don't recommend fighting them back (the above screen), but quickly putting down your weapons, surrendering or agreeing to go to prison. Otherwise, you'll be surrounded by a large group of highly trained enemies who are very hard to beat.