How to get a second kingdom

Here's how it works: If you are their liege, the gains from the war are part of your kingdom. It doesn't matter if they're your king, duke, earl, or even mayor, bishop, or count.

To claim a duchy from somebody else through war (when you don't have a casus belli that lets you claim it directly):

1. Find out who* has a claim on it by opening the duchy information screen and clicking on "Claimants" to see who has a claim on that kingdom.

2. Invite one of the male claimants over to your court. (If nobody will come over, or there are no male heirs, you'll have to find another way to get that duchy. Note that if one of your courtiers marries a female claimant that you can't legally give a title to, that female claimant is not your vassal and the newly-conquered duchy will not become part of your kingdom, even if you give your male courtier a title.)

3. Give that claimant a landed title. Any landed title will do, even a city, church, or castle.

   Note that if you land the claimant a city, gaining him a Duchy will found a vassal merchant republic instead of gaining you a new feudal vassal if you are playing version 1.9 of the game or later. Merchant republics are generally considered awesome to have as vassals due to their revenue and tax bonuses, but this might disrupt an existing merchant republic that is already earning you revenue.

4. Go to war with the ruler of your choice, using the "Claim on duchy X" as your casus belli.

5. Once you win the war, that claimant will be your duke.

I'm guessing that this will also work with kingdoms if you're an emperor. So far, I've claimed two duchies this way: once by giving away a county to the claimant, and once by giving away a bishopric.

*As revealed in this other answer, you can't actually press the claim in war unless the claim is strong, or the claimant is a pretender (2nd or 3rd in line), or there's already another war going on over the title, or the current holder of the title is a woman (and you have a male claimant) or a child (so there's a regency). You can only use "any claimant with a pulse" if the title is disputed or the ruler is a child; you can only use "any male claimant with a pulse" if the ruler is female; otherwise, you'll need to pick up one of the pretenders or someone with a strong claim: you'll need to do slightly more research to find who they are, and they're often harder to invite to your court.

(Note: if there are three or fewer male claimants, then it's almost certain that they're the heir and pretenders. Just grab one and go on your way. Figuring out the exact claims is more important when there are four or more male claimants.)

King can never be a vassal of another king, and there is no “partial vassal” or something like that (and the same applies to all the other levels too).

The duchy will become part of your kingdom (de facto, this won't change which kingdom it is part of de jure). But if the crown authority in the other kingdom is high enough, you won't get the duchy (it prevents titles to pass out of the realm through inheritance).