Hurmmm, Well first off i'd recomend PJIRC, i'm not sure how much bandwidth load it takes up... Most websites can normaly handel it unless you decide to have.. a TON of users connecting via the applet. I'd recomend premoting the use of the IRC channel you're using, along with mirc, or another type of script or client... Also maybe offer a few "how to get on IRC guides" for differnt clients...
As for a server, you need to hook up with a server that offers a java applet, you can't just slap one on and call it good. Our network uses pjirc, for most of its java stuff, but we are having a few issues with some of the major channels lately. I'm taking a look at one website tonight, its costed us quite a bit of user flow, plus the website was down for about a week, now the applet itself is "failing" but i think someone touched some of the files we use, probably deleted them.
PJIRC *may* work with out being directly connected or affiliated with a network, but from my knowlege you need to find a IRC server that offers the applet.
Dedicated servers are can now run IRC servers(Last i looked in the TOS that is.) Shared cannot though.
Anyway some helpful links...
PJIRC
mIRC
Uhm, a side note, it says PJIRC can run "stand alone" so it should work without having to find a network to Host you.