When you reach the threshold previously required to attract the king of your civilization, a new option would happen.
You make a choice: Either accept the king into your fortress and remain part of your civiliation (Same functionality as it exists now) OR declare independence and raise your own noble up to king status, making your fort a brand new dwarven nation.
If you choose the 'independence' route, your old civiliation would temporarily become hostile to you. After a period of a year or two, they would send a single very well armed very well armored siege against you. Their goal is not the destruction of your fortress but the death of your primary noble. Loyalty cascades would be avoided because your dwarves would be a different civilization at this point, and thus it wouldn't be seen as 'attacking your own'.
The dwarven king would be part of the attacking siege. The siege immediately ends the moment either your noble dies or the dwarven king dies.
If your noble dies, the siege is called off and all the dwarven troops leave and your fotress is returned to ownership by your parent civ. However, you still lose the dwarves as a trading partner.
If the king dies, the siege retreats and you are declared victorious. The dwarven civilization will get a new king and be no longer 'hostile' to you. Meanwhile, your noble is promoted to the level of king. In this case, you WOULD be able to trade with your former dwarven parent civ even once you reach king status.
If you happen to have embarked belonging to a completely dead civ, no independence war would be needed. Your noble would instantly be crowned king and your civilization would no longer be seen as dead. The same would happen if you found your fort on an island with no other connections to other dwarven civs.
As the king of an independent nation, you would have the option of sending 7-dwarf expedition forces out to found new forts. You would have to provide the dwarves and give them equipment and supplies. Their success would depend on how well trained they are and the nature of their starting equipment. Forts that are successfully established would become new trading partners and gain access to materials and goods depending on the location of the area they colonized.