iPhone Broken???

Make an appointment at the nearest Apple store to have your iPhone evaluated.

Once the diagnostics are complete, the staff will explain your options.

Make a Genius Bar Reservation

http://www.apple.com/retail/geniusbar/

Your phone is out of warranty and, in that case, Apple offers an attractive OOW swap for a refurbished unit. You will, however, lose all of your date. If you have not backed up your data, you might wish to deal with an independent repair shop. If you do, make certain that they will assume the risk in the event that Apple subsequently denies an OOW replacement. Below is a policy from one such shop.

We will be responsible for maintaining your eligibility for Out of Warranty Swap at Apple for iPhones for US customers. If we accept your device for repair and it would have been eligible for OOW swap, but you choose to send it to us instead of simply going to Apple and paying the Out of Warranty Swap fee (see Apple Support for those prices), then we will return it in a condition that will allow it to still be eligible for Out of Warranty Swap by Apple. If you receive your device back from us for a no fix repair, and the device is now deemed ineligible for Apple OOW Swap, then you can send it to us with the Swap fee and we will swap the device for you and send you your new phone as part of our customer service.

Philly_Phan wrote:

Your phone is out of warranty

How do you know that? The 5C is still available for sale. Verizon, for example, still stocks limited quantities.

I've just learned that your phone may be in warranty. If it is and you have a backup for your data (or don't care about your data), it's a no-brainer - take the phone to Apple. Otherwise, look for an independent repair shop that has a policy that protects you for warranty repairs just like the OOW protection that I described previously.