The best way to achieve success with a newfound kingdom is 90% preparation. I always prefer to accept a vassalhood, get a castle and build a stockpile of reserve troops before I break away. Even if it isn't in a good location, having those pretrained troops ready to shuttle over to your choice of starting location when the time finally comes is a godsend, and having a disposable castle to draw besieging armies toward can buy you precious time to deal with other threats if multiple factions decide to crush your upstart arse.
This is my rough rule-of-thumb checklist:
- 200,000 denars minimum
A new kingdom is expensive, and you need to be able to absorb a lot of financial loss to see you through until you can force your enemies into peace or at least gain enough breathing space to be able to start upgrading your infrastructure
- 175+ troopcap
I feel this is enough to beat almost any lord in a 1-on-1 fight with few losses, and that's something you need to be able to do in order to eliminate solo operators when/if you manage to break up the big marshal-led armies
- High training skill and village rep in your planned starting area
This allows you to recruit as many... recruits... as possible per opportunity, either passively via your steward or by going from village to village. Aside from the obvious need to replace your combat losses, it also serves you well in creating a quick and efficient training mill system to augment your vassal's own self-raised forces
- 20+ rep with all good personality lords of factions I plan to begin within/near
This should be enough to flip most lords over to your side, and being able to do that to a half dozen of your enemy's vassals in a short span of time not only increases your own kingdom's strength and size but weakens your enemy at the same time (I always play with lords bringing their holdings with them when they change allegiance.)
- 30+ Right to Rule
This is essential in establishing your kingdom's legitimacy and security. Even if you manage to survive the initial onslaught, the other kings need to you see you as a peer and not just an upstart. If they don't, they can and will stay at war with you for a looooong time and declare war without pretext. Periods of peace are important for growing your economy and dealing with minor factions, rebuilding lost troops etc. It never hurts to have good rep with the other kings either as it makes them more willing to sue for peace and recognize your claim.
That's all I got atm, hope it helps! Oh, and I should mention that I haven't gotten to the kingdom stage in 3.6 yet, but from what I've heard around the forums hiring a strong merc company as soon as you go indie can really take the edge off of those rough early days.