Well, your cable should be covered under the 1 year warranty that comes with the iPhone itself (unless you purchased a refurbished unit then you have a much shorter warranty period closer to 90 days). Since you say your iPhone is only a couple months old, I would think you would be best off to take your cable, charger, and iPhone to the Apple store and have them go over it with you to determine what is wrong.
In the mean time, if you know someone else who has an iPhone 5, you can try to charge their phone with your cable, and try to charge your phone with their cable. This should tell you if the problem is with the cable or iPhone (or both). I do know that the iPhone 5 lightning cables are not as robust as I would have hoped. My wife kills them in about 2-3 months time, but that is because she puts heavy stress on them by making a sharp bend in the cable right where it comes out from the connector. The cable first starts to show a bulge right there, then it starts to split open and disconnect/stop charging. I have had 4-5 replaced at my local Apple store since the beginning of this year.
If you call AppleCare on the telephone, they should be able to send you a replacement cable, but you may have to send yours in first. Based on that, you may be just as well off to buy another, and use it until you can make it to the Apple store yourself. Around here, you can walk in with the dead cable and walk out with the replacement 10 minutes later (on a slow day). I find that having the spare cable is nice so that if something happens to one, you can still use your iPhone until you get it replaced, and at the $30 price it's not too excessively priced for "assurance".
