I think you have three basic options...
1) If your existing router supports standards for relay/remote on 802.11g networks, the Express will probably be able to "extend" the network. What this means is that you continue to have one wireless network, and the Airport Express has access to both the internet and the intranet without having to be physically connected to the other router. In some cases you might not be able to extend, but you can still "join," meaning that the Express has access to the intranet, but devices cannot join the network via the Express, and it does not function to extend the range of the network.
2) Otherwise, the next option is to plug the AE into a wired port of the router, which places it on the subnet. From there, it will create a separate network, and you can use the separate network for airtunes.
3) The final option is to have the AE not plugged into or joined to the router. It will not have access to the intranet, but you can still access the separate network that it creates solely for the purpose of Airtunes broadcasting.