The Dreamcast is probably one of the single most hacked consoles ever made. It may even beat out the original Xbox, which people have done all manner of things to. There should be some sort of adapter or method for installing an ethernet controller onto the console, though you may have to resort to places like ebay to find the part.

Beyond that, I wouldn't go getting my hopes up too much. You can run Linux and NetBSD on the Dreamcast, if you get a keyboard for it, but I don't really think there are any native web browsers for it. You'd probably be looking at text mode only browsers, simply due to the limited hardware resources of the console.

I'd use Google to find some Dreamcast fan sites. You're likely to find far more comprehensive info than the bits and pieces I can dredge up from memory.