How do I get a castle?

The best way of getting a castle or town is to capture one yourself. Keep an eye out along the borders where towns and castles frequently change hands. Those places usually have a rather weak garrison so you should be able to take it without too much trouble.

When you capture a town or castle, you can make the request to your king that you want to keep it. Unless you've just recently been rewarded a fief, the King will almost always grant you the town/castle you've just captured. Don't count on the king giving you one otherwise unless your reputation with him and the other lords are all super high and you manage to convince them to support your cause.

If the castle near your village is already owned by a lord (almost 100% of the time it is), then you are out of luck, you can't get that one unless an enemy faction captures it and you retake it.

To quote the wiki:

Castles are found all over Calradia, and each one has a nearby village. In order to take control of a castle, you must besiege it...

With an enemy castle...you have the option to besiege it and claim it for your own...

There are other ways for a faction to capture a castle in Warband other than just besieging it. They can also wait for the food supply of the castle/town to run out, and then take control, or if a lord holding a castle/town defects without being indicted, it will go to the new faction defected to...

Should the player conquer a castle without being sworn to a liege, he takes the castle for himself, and founds a new kingdom...

If the player conquers a castle when they already have their own new kingdom, they are given a list of Vassals that are with the player's kingdom to take control of the castle, the option for the player to take control of it is available, and the option for the castle to have no Lord is available as well.

So, you can either revolt against your current kingdom to found your own and conquer a castle for yourself through one of the methods mentioned (or conquer and then possibly decide to revolt if the king doesn't give it to you) or become buds with your king and convince them to appoint you as lord of a castle.

If you conquer a castle yourself, like when conquering a village or fief, you can ask your king to give it to you and if they don't you can either renounce your oath and fiefs, renounce your oath alone and take your fiefs and the one you just captured thus causing you to rebel, or you can take a payout. The kings apparently make the decision randomly, but key factors seem to be renown and relationship with the king although there is some mention that each fief, city and castle holds some value and that kings try to balance this value out across all vassals. Apparently you can possibly convince other lords to support your claim to the castle with some persuasion.

The more the king likes you the more chances you have of him giving you the castle you conquererd. In my time i had 3 little fiefs, no castle and no large city. When i finally captured my first castle the king said he would prefer it to give it to his friend instead of me. But because i was a nobleman i let it slide, now the fun part was that about a few moves after the castle was again under enemy control. So then it was the second time i conquered the castle alone (This time was easyer because there were only recruits present in the castle, and i gave them free passage if they left) . Now i asked it again, and because he already gave it as a present to another "friend", he was able to give it to me.

I would suggest trying to capture one alone and you should have a higher percentage to get it. (I have no source for this, but I did capture a great city alone and the king gave it to me directly)