Not able to charge iPhone 5

Solution 1:

Some iPhone 5 are eligible for a free battery replacement. Maybe that is related to your issue?

Apple has determined that a very small percentage of iPhone 5 devices may suddenly experience shorter battery life or need to be charged more frequently. The affected iPhone 5 devices were sold between September 2012 and January 2013 and fall within a limited serial number range.

You can test if your phone is eligible, on this website.

Solution 2:

Before checking for a battery replacement, I would clean the connector of the iPhone. Check this video (I will also work for an iPhone 5)

I had this problem and it solved it. If this doesn't work, go with jherran's answer.

Good luck