Have you tried restoring it or resetting it to see if that helps? Might be a bad cable...try troubleshooting on Apple.com/support and if you still can't resolve the issue contact Apple (or if there is an Apple store near you then you might want to stop by there).
Hopefully it's nothing serious! I haven't had any problems with it charging. I used my old iPhone wall charger though and just put the new cable on it, but I doubt that did anything special because the OP mentioned they were having trouble with it charging and being recognized via the computer also. Hopefully it's just a bad cable and not the port. But I would suggest going to the Apple store if you have one in your area versus the VZW store for hardware issues. I personally would have to go 100 miles for the nearest Apple store so usually if I must go in for something I would likely go to VZW (after trying Apple support online and over the phone lol)...just because getting help from the genius bar tends to be more successful than VZW. Not ALWAYS the case but more often than not, it is.
We want you to have the best experience with your iPhone 5. Let's check a couple things to verify where the trouble is taking place. Are you using the older chargers with the adapter or just the new cables from the iPhone 5? Since you mentioned connecting it to your Macbook Pro you want to make sure that when connecting to a computer to gain a charge, the iPhone needs to be powered on. If the phone has died due to low battery then only the wall charger will charge the device. Please keep us posted to get this fully fixed for you.
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Mine won't charge either, my daughter spends 15 minutes every other night trying to get the cord in "tight" enough. This last time nothing works and its about to die AGAIN. What do we do?
Is there a reason that cord wasn't fitting tight enough (and is it not fitting well in the phone or the wall charger)? Is she charging it with a case on? You might need to have your device checked out. I still have yet to experience any problems on either end. I have a case on it but I don't have any issues charging with it on either. Hope you are able to find out what the problem is.
Make me Wonder if it's in the L.W. Connector with it Being new and All. And when you mentioned trying to get it in tight in the Socket if the pins are of the Fragile type something might have broke off like one of the connecting Pins..
I just picked up my 5 & I am having the same problem. I even purchased a second lightning wire & usb charger for the car. Neither wire seems to fit snugly. Is there something I'm missing?