As mentioned by @Daft Daemon Prince is the most powerful a mortal can hope to become. There is sort of an official source (though soon to be defunct) the pre-8th edition Chaos Space Marine codex talks about how being raised to a Daemon Prince is the "Ultimate ambition" for most champions of Chaos and if they further the cause of their master they are given the "precious gift of Daemonhood" and can serve their master for eternity.
The most infamous of these Daemon princes are the Daemon Primarchs.
Greater Daemons on the other hand are fragments of the Chaos Gods themselves so manifestations of their warp power and aren't created from 'mortals'
The Chaos gods ultimately come around from 'thoughts' and emotions this is why the Emperor was so keen on spreading the Imperial Truth and trying to remove the use of the warp for travel. He wanted to remove human's reliance on it and starve the Chaos gods of their power.
I don't know if this ultimately would have 'killed' the Chaos gods but it certainly would have weakened them. The reason I don't think they can be killed is that even actions seem to create warp entities
If you've read the Horus Heresy novel "Master of Mankind" they do explore this a bit more though is obviously still reasonably vague on the nature of the warp but,
we know that the first murder ever committed in cold blood actually creates a daemon manifestation in the warp that can never be killed - due to the emotions involved. So it is thoughts and emotions that create this.
Ultimately the Emperor wanted to remove the reliance on the warp so he could better manage/guide the emergence of human psykers so that the Warp wasn't as big a risk to mankind