It isn't going to be the same as in medieval Europe, that much we know. From what's been described, succession laws are a simple matter of the Clan leader nominating a successor from their family - and that's it.
The major factions or kingdoms are composed of a number of different Clans (think of them as noble families). The kingdom is ruled by the leader of one of those families. The other Clans compete with each other for position within the kingdom, and ultimately for the opportunity to replace the incumbent Clan as rulers of the kingdom. Quite how this comes about hasn't been described in detail, as far as I'm aware, but it's likely to be something to do with building up your own Clan's renown and influence, as well as strategic relations with the other Clans in your faction, so that you can make a challenge for rule when the time is right.
I'm assuming that when you marry your player character to some NPC, whoever they are and whatever their rank and position, that that NPC becomes a member of your own Clan afterwards, and any offspring you have are also in your Clan. I don't see how the system could work, otherwise.
But whatever the details, the point is that you absolutely can join an existing major faction and work your way up to become the ruler of that faction, unlike in Warband. The precise methods and processes for doing this will become clear once the Early Access comes out in a few weeks' time.