I believe when you make a claim you form the equivalent of your own city government (which I think is called a "site entity"). The parent civ is the civilisation that you belong to. This doesn't mean that you rule the territory, but you can see the entity in legends mode under "Civilisations and Other Entities". If you claim a site and manage to wait until a 10 year boundary is crossed, you can even see the influence for your home civ being placed on the "Historical Maps" in legends mode. However, if you ask the surrounding populous about the ruler, they are likely to *still* refer to the previous civ -- that's because the site does not change civilisation hands unless it is destroyed.
To destroy the site (make it a ruin), you must kill all of the government officials in the town. However, as previously stated, this will usually just result in a new government official being nominated. You can keep killing them off and eventually they will have to send an expedition to take over the ruins. It is possible (but unfortunately unlikely) that your civ will beat them to the punch -- because dwarfs are notoriously slow about that kind of thing. Goblins, on the other hand, are pretty opportunistic and I've had a few gobbos take over ruins that I created.
Interestingly, while the city is a ruin, you can hand over the title of lord to one of your companions (I often conscript one of the locals just for this purpose). Then you can run around spreading the rumour of the crowning of the new lord. Eventually, all of the people in the area will claim that this person is the ruler -- however, it doesn't make the town un-ruined and only lasts until some expedition comes and reclaims the site. The one thing I haven't tested (and has been on my TODO list for a while) is once you've won over the whole populace, and the ruined site gets reclaimed by a different civ, is it easier to start an insurrection? Not that it really does any good, but it might be interesting.
Essentially this stuff is unfinished, and it is practically impossible to get in a situation where you are the actual ruler. However, you can mess around a fair amount and generate some interesting situations. If you are waging a personal war against a civ, ruining their towns over and over again is a pretty effective way of hurting them.