Each of the six factions has a claimant to its throne, a noble who believes they are the rightful ruler. You can find these claimants in taverns once you have sufficient renown. Speaking to them will initiate a questline where you pledge your support to their cause. This involves gathering support from other lords, capturing castles and towns for your claimant, and ultimately defeating the current king.
Supporting a claimant can be easier than starting your own kingdom, as you leverage an existing noble's claim and potentially gain allies more readily. However, you will still be a vassal under the new king, albeit a very influential one. The challenge lies in convincing other lords to defect to your claimant's side and fighting a civil war against the established monarch.