First step: be sure networking is OK.
In your case I would say:
1) LAN cable in HP Win7, setup home network.
2) then hook up Dell XP on cable , setup connect to home network.
Check if PCs see eachother, as long your LAN isn't correctly setup all efforts to multiplay are futile.
Use HP W7 as duke server
Use Dell XP as client.
(setup a LAN server/client in eduke is self explanatory, follow menus)
Be sure eduke is allowed to pass through firewall.
LAN = ethernet in your case, do you need that router? (Can block, maybe you need port forwarding? try directly between PCs first if you can)
[Edit]
Has been a while, checked settings and eduke32 server/client
To launch eduke32 server:
eduke32 -server (easiest is to create shortcut and adapt command)
All options for multiplay will be at your fingertips ( be sure to allow firewall access!!!!)
-> Start game
To connect client via LAN:
eduke32 -connect *server ip*
Same, easiest is to create extra shortcut and adapt launch command.
Same again: firewall access, if you use a router between PCs maybe you'll have to forward ports (not sure ....., depends, read router manual)
[Editē]
Networking OK (Win7 PC)
Launching eduke32 server (Win7 PC)
Game server launched (Win7 PC)
Client connected to eduke32 server from XP PC
shake them baby !!!