Scattered around Skyrim in ways that may or may not make sense are both the remnants of the ancient Dwemer civilization and the pieces of equipment used and inspired by them. It is these relics that Calcelmo, the Court Wizard of Markarth is focused on, and it is through this that you may get his attention without ever nearing the capital of the Reach.
Should you find a Dwarven Greatsword at any point and use it, you will find that shortly you receive a letter via courier from Calcelmo. The wizard expresses through the note his interest in Dwemer remnants and asks that you bring him the sword so that he may take and study it. When you arrive in Markarth and get to Understone Keep, you can find him by heading to the left at the first opportunity, where you will find the wizard and his assistant, across a stone bridge from the entrance to a Dwemer ruin.
When you give him the sword, he begins to view you favorably. As an alternative for gaining the wizard’s favor, you can perform some other service for him- such as bringing him the ring he requested from the silversmith stall in the outdoor market of Markarth, or helping him in his quest to win the love of Faleen as a part of your work for the Aedra known as Mara. Whichever way, you can convince him to grant you admittance to Nchuand-Zel, the ancient ruin underneath Understone Keep, on the condition that you help him in turn- there is a huge spider by the name of Nimhe occupying the excavation area above Nchuand-Zel proper, and your entrance to the ruin is contingent on your promise to slay the immense vermin that is preventing Calcelmo’s excavators from digging up more relics for him.
As a nice side bonus for slaying Nimhe, Calcelmo also offers to let you have free admission to the Dwemer Museum he runs, which is on the opposite side of Understone Keep from the ruins and excavation. Of course, once you have access to Nchuand-Zel, there’s little reason not to fully explore it- just keep in mind that the dungeon used to be an entire Dwemer city, and thus is especially large. If you prefer, you can return multiple times, but it may wind up easier to recall where you have and have not been if you simply make it a single large expedition, emptying your inventory of absolutely everything not neccessary and bringing along a Follower as a form of pack mule (in addition to combat assistant). What’s more, the place is full to the gills with smeltable Dwemer scrap, so anyone looking to improve their skill rank in Smithing will love this place absolutely to bits, as it provides an absurd amount of raw material with which to craft arms and armor, and thereby gain ludicrous amounts of Smithing experience.