(Edit: I've been assuming you and your cousin are on the same network since you said you were using tcp/ip. If that isn't the case then yes you will probably need to go through a server or some other third party program. I haven't used it but you might be able to just install GOG Galaxy and go through it. If that doesn't work or you don't want to use it for whatever reason you could try something like hamachi or tunngle. I've never used tunngle and didn't care for hamachi so I'm afraid I can't help you there.)
I usually go to network places and then in the bar at the top, to the left of the search bar, type \\"computername". ( Make sure that those are backslashes.) If the computer you're trying to access has a password you'll be prompted to enter the username and password. Here's more information on that if that wasn't very clear.
http://windows.microsoft.com/en-us/windows-vista/find-files-and-folders-on-other-computers-on-a-network If that doesn't work I can remember a few things that might cause trouble. I suppose first of all you might make sure it isn't someone's antivirus software blocking the game. It should be pretty easy to whitelist the exe of battlecry, depending on whatever antivirus you might be using. Turning on network discovery can also help sometimes. If one of you is using windows XP and the other a newer version of windows (7 at least not sure about Vista.) then you're in different workgroups and won't be able to connect. I think that link talks a bit about workgroups though. Other than that it might be a firewall. If it's a firewall you'd probably only need to, again, whitelist the exe or something.
I can't think of any other common problems and I don't think there's any specific trick to getting Battlecry 3 working. If there's anything that needs more explanation Google or whatever would probably be more clear than I can, but I'll answer anything as best as I can.