This has nothing to do with backtrack, but rather basic networking skills.
The only problem I see here is that your internet connected network can't find an appropriate DNS server. It would be a good idea to find out what the router thinksiis the gateway and ensuring that this is correct. Generally for a home user type scenario, the gateway is the public facing IP address assigned by your ISP.
Accessing both networks is a trivial matter of ensuring that the responsible router has the gateway's in it's tables for both networks, the client (NIC) tries to access.