I would first try letting it charge for about 10 minutes. If the phone is completely dead, it may take a little time to give it enough juice to start up.
However, I'm assuming you've already tried that, and it seems like too easy a solution, so next is to try doing a soft reset. Do this by holding the home + power button at the same time for about 15 seconds. If you are lucky, then the Apple logo will appear and once it boots up fully, hopefully it will charge as well. That is the simplest solution without having to get your hands dirty.
If neither of those steps work, then you may have to open up the iPhone. If you do not feel comfortable doing this, bring it into a local cellular repair shop or sell your device on eBay, Craigslist, etc. or if it's a relatively new purchase, bring it to Apple. They can at least tell you what may possibly be the problem, and point you in the right direction.
But if you do open up the iPhone, check the battery, lightning flex cable. Hopefully it will be one of those two that have gone bad, otherwise you're looking at repairs on the motherboard, which may not even be worth the hassle.
Best of luck!