I would start with having the battery tested. There are jeeps in much colder area's that start and run fine. I had problems with my battery and had it replaced under warranty....

 

If it just clicks and is not turning over it has nothing to do with the fuel pump/filter.. If it was turning over and not starting that would possibly be the fuel pump. If it eventually starts without a jump or charge, I'd venture to say the starter solenoid* is sticking.. Then u keep trying and eventually it turns over. Next time it does it tap on the starter solenoid. I bet it starts right up

 

I had trouble starting the Srt yesterday in the morning. It clicked, clicked and clicked and then while turning the key, I pressed the gas and it started up. Last time I checked the srt's were efi, so wtf gives? Lol

 

Like the others said, start by testing the battery. Cold weather takes a toll on it. If that is in the clear next guess would be the starter.

What year, are you still under warranty?

 

its been ****ing cold here in NY and I havent had any issues with either the key to start or the remote starter. I have on 08...sounds like the battery

 

Sounds like a starter problem.