To be accepted vassal you need renown (probably single most important attribute in game). 150-200 should be enough. Kings can offer you vassalage even on their own if it is that high. To get castle or town you need to have renown higher than any other homeless lord in kingdom. This means renown around 500-800 usually. Of course there have to be unassigned castle in your kingdom, which happens when kingdom captures castle from other kingdom. If you take that castle or town on your own, that helps little bit as well. Renown is gained by winning tournaments and difficult battles (especially those where you are outnumbered). You can also gain renown by hosting feasts but for that you need to be married and have a castle or town already.
To get married you have to win favour with lady or its father/brother. Usually both. Free ladies can be met at feasts and best way to get going is to win tournament which is held at the same time as feast, then dedicate victory to lady. Then just tell her you admire her and she will let you know if she wishes to meet you from time to time. You can get permission from her father/brother to meet her, else you can meet her just in discrete. Once relation with her is high enough (20-30), you can ask her if she think you can have common future. If she say yes, You have to gain permission from her father or brother. For which you need good relation with him. That is just general and most common way, there are other depending on character of lady and your standing (your renown, ownership of fiefs, honour and so on). Oh, and usually you need quit big sum of money too

that marriage gona cost you something. Bards and poets in taverns will tell you lot about this topic.
To help claimant, you need to find one (just ask traveller about their location) and agree to help him.
To start your own kingdom you need to conquer castle or town wile you are not vassal of anybody or to cancel your current vassalage while keeping fiefs you obtained from king. You want to have high Right to Rule or else it will be difficult to keep your kingdom.
In general you want to develop your character little bit before starting pursuing some of those goals. Just wander around, fight bandits, rise soldiers, win tournaments, get better equipment, recruit companions, gain relations with lords, kings, villages and towns by doing quests for them.