You need first wage war against the faction and start attacking the castle. The faction that the castle belongs to will also be your enemies. Make sure the castle you are attacking will be your enemies for the rest of the game. The second option is to siege a small castle at the early stage of the game. To have your own independent settlement, you should have your clan rank 3 and 100 troops with you. If you choose to give it to Arzagos, you can have your own independent non-imperial settlement. If you choose to give the dragon banner to Istiana, you will also have an option form an imperial independent settlement. After that you will have the option to join the forces with Empire or against the empire. After getting the banner you can present it to Istiana or Arzagos. After completing the Neretzes Folly quest, you will be asked to retrieve the dragon banner from the hideout. The first option is to play through the main quest. And there are two ways in which you can set up your own settlement. It is difficult to own your settlement at the beginning of the game. Settlement in this game is basically your own town or castle. To make a faction your enemy you need to take hostile action against their villages. You can only siege a town if they are your enemy. Note: To have your own kingdom, you need to siege a town.

But there is also another way in which you can form your own settlement at the beginning of the game. In Mount and Blade Bannerlord, players have to reach a certain stage of the game in the main quest to form their own settlement.

Mount and blade how to get a castle