When I start my 2002 Chrysler Concorde there is a strong gas fume coming from the exhaust. Does this mean I need a fuel filter? If so, what is the typical cost for getting a fuel filter on this kind of car.

You are looking in the wrong place. If the fuel filter was obstructed, it would deprive the engine of fuel and therefore less fuel would be delivered causing a lean condition. Smelling raw fuel at the exhaust tells you that either too much fuel is entering or it's entering at the wrong time. As nearly every sensor on the engine plays a part in fuel delivery, I can't give you an accurate diagnosis but instead, i recommend that you have the computer codes read (most larger parts stores do that free) and see if you have any stored malfunction codes. Only item that may not produce a code that could have something to do with your problem is the fuel pressure regulator. If that fails it can cause a rich condition by over pressurizing the injectors. An exhaust leak ahead of the O2 sensors or a vacuum leak at the engine can do that also but usually not to the point that you smell fuel.
If you have not replaced the filter in the past three years, you can replace it, but I'd do that after you find the actual problem. Most times the filter is under $25, but labor varies from shop to shop.