Forget the fuel filter. On Chrysler products they typically last the life of the vehicle. The pressure isn't critical either like it is on some other vehicles. You will hear the fuel pump run for one second after turning on the ignition switch. That's where the pressure you have now is coming from.
The first thing to do is to check for spark. My bet is it's missing. That means the fuel pump will not turn on again during engine cranking. Those two circuits are tied together so you can't get hung up on the first thing you find missing. For no-spark, especially when it started out as a heat-related intermittent problem, suspect the crankshaft position sensor or the camshaft position sensor. Most of the time there will be a diagnostic fault code related to the failing sensor. I'm racing a dying laptop battery, so rather than retype everything, go to this page to see how to read the codes. Chrysler makes reading them yourself much easier than any other manufacturer. The only difference is instead of counting the flashes of the Check Engine light, on your car the code numbers will show up in the odometer display.
Once you get the code numbers, go to this page to see what they mean:
Monday, May 5th, 2014 AT 9:00 PM