I have never run a server behind a router with private IP addresses, but it is my understanding that you will need to configure your router to direct QTSS traffic sent to it to 192.168.0.3 or whatever the private IP of the streaming server is by setting up port forwarding.
Requests from the outside sent to the router on ports 554, 1220, 7070, 8000 and 8001 (the ports QTSS uses...) need to be directed to the server otherwise they will just be stopped.
My NetGear wireless router can work with DynDNS.org and other dynamic DNS services. If yours has this feature, use it to give your streaming server an unchanging host name (something.dyndns.org, etc....) that will consistently point to your dynamic IP address.
Keep in mind that many ISPs don't allow you to run servers in your service agreement. If yours doesn't, you may run the risk of being terminated if they find out. I would put access restrictions on the streams to limit their distribution to authorized users only. This will limit the bandwidth consumed by serving your streams and draw less attention from the powers that be.
Good luck,
Peter