BNC & CAT5 Help

Hi All

I need help with a client of mine.

They are running a DOS accounting program with a Netware file server all connected via BNC cabling.
Now they also have a seperate machine that connects to a router for DSL internet which is connected via CAT5 cabing.
Now they want to connect 2 of pc's which are on the BNC network to connect to the router for internet access.
My plan of attack was to install seperate NIC'c in these 2 pc's and run the cabling to the router in CAT5.
I'm not sure if this will cause a conflict?
How do I go about making the browser go to the correct NIC for internet access?

Any suggestion would be greatly appreciated.