It is possible to play online on a PlayStation 2 Slim, and if that's the case, how?
I would personally try a LAN connection with an Ethernet cable. Let me know if that does not work.
Do that by following these steps: 1. Turn the PS2 and broadband modem off. 2. Connect the network cable to the modem’s Ethernet port and the Ethernet port on your PS2 network adapter. 3. Turn the broadband modem on. Now turn on the PS2. 4. Load the Network Access Disc in the PS2. 5. Follow the on-screen setup instructions. Refer to the Network Access Disc manual for further support.
last I had heard the servers for the ps2 games are all shut down (much like wii) and online play for games is substantially impossible unless you are doing this on a moded console and have access to a third party server but this may need some challenging home networking if these servers still exist.
there is also the possibility to pair consoles for a system link with ethernet I believe but game support for this would have to be enabled.
homebrew also offers some custom bbnetwork if I recall correctly but I doubt it still running.
homebrew does enable ftp streaming of games so if you are technically inclined you could establish a network with your pc and host files to it so the ps2 is essentially streaming your game to your console over your private network.