What you want to do can be done. However you would probably want to use a separate "deep-cycle" battery instead of your car battery. Car batteries are not really designed to have a load placed on them for extended times but rather short bursts of load, such as starting your car.
With that said. A deep cycle battery would do fine for low wattage, long time drain but as you increase your wattage you would probably want to have your alternator supply the power.
This can easily be accomplished by running a line from your alternator to your deep cycle battery (s) with a toggle switch placed in line so as to allow you to toggle your power/charging on/off as needed.
Check to see what your alternator output is rated for. To make sure it can handle what ever load you will be placing it under. Make sure your inverter can also handle the load. And determine how much power you will need so as you know how many batteries you should have.
I used to be a truck driver and used this set up to run a small refridgerator, microwave, tv. playstation and satelite tv system.