ok first off i would like to say that i do not reccomend PS2 online and xbox live is way better. but, if you insist on doing so here's how ya do it.
I set mine up a while ago but i'll try and...refresh my memory:
go buy a ps2 network adapter (plugs into the expansion bay in the back)
if you have wireless internet, you need a reciever of some sort.
whether it's wireless or wired, you will plug your ethernet cable into the the adapter.
it comes with a start-up disc and you put that in your PS2 and follow the directions it gives you
it will auto-detect your connection (IP address, DNS, etc.) hopefully
if auto-detect doesn't work, then go to your PC which is already hooked up and find out your connection details and type them in.
that shold get you hooked up
if you have a new, small ps2, then you just plug your ethernet cable into the back of the system (correct me if i'm wrong)
have fun and reply back if you get it to work