iPhone 5s won't charge
We'll need to troubleshoot the charging method first - then the device itself.

Troubleshooting the charging method, adapter, cable:


Compare packaging and writing on the cable:


  To identify counterfeit or uncertified cables and accessories, look carefully at the accessory's packaging and at the accessory itself. Certified third-party accessories have the MFi logo on their packaging. An Apple Lightning to USB cable has "Designed by Apple in California" and either "Assembled in China," "Assembled in Vietnam," or "Indústria Brasileira" on the cable about seven inches from the USB connector. You'll see a 12-digit serial number at the end of this text.

Compare connectors and laser etchings:


  You can use the Lightning connector, USB connector, and laser etchings to identify counterfeit or uncertified Lightning accessories. Compare the images below of certified Apple accessories and counterfeit or uncertified Lightning accessories. These are only a few counterfeit products. There are others.

Lightning to USB cable, Lightning-connector end:

The Apple-certified will have:


  
  Single piece Smooth Rounded, smooth contacts - (Image)
  Width and length of Apple boot is consistent (7.7 mm x 12 mm) - (Image)
  Gray/metallic faceplate insert - (Image)
  


vs. counterfeit or uncertified will have:


  
  More than one piece Rough or inconsistent finish Squared contacts with an uneven surface - (Image)
  Varying width, length, or thickness - (Image)
  White or black faceplate insert - (Image)
  

Lightning to USB cable, USB-connector end:

The Apple-certified:


  
  Interlocks on USB shell are trapezoidal Interlocks are equally spaced from edge - (Image)
  USB contacts are gold-plated - (Image)
  USB surface is smooth and consistent Surface of USB shell is flat - (Image)
  Surface of the insulator for the USB connection is uniform and flat - (Image)
  


vs. counterfeit or uncertified:


  
  USB shell interlocks have right angles Interlocks are inconsistently spaced - (Image 1 Image 2)
  USB contacts are silver-plated - (Image 1 Image 2)
  USB surface is rough or grainy Surface of USB shell has a detent or retaining divot - (Image 1 Image 2)
  Insulator for USB connection has notches or indents - (Image 1 Image 2)
  



Source: Apple

Troubleshooting the device itself:

Plug into power by one of the below methods:


Plug your charging cable into a USB wall adapter, then plug the adapter into the wall.
Plug your charging cable into a USB 2.0 or 3.0 port on a computer that's on and not in sleep mode. Don’t use the USB ports on your keyboard.
Plug your cable into a powered USB hub, docking station, or other Apple-certified accessory.


If your device won’t charge or charges slowly:

If you have very low battery and plug into power, a black screen or a black screen with a red battery might appear for a few minutes. Follow these steps and try again after each:


Check your charging cable and USB adapter for signs of damage,* like breakage or bent prongs. Don't use damaged accessories.
Use a wall power outlet and check for firm connections between your charging cable, USB wall adapter, and wall outlet, or try a different outlet.
Remove any debris from the charging port on the bottom of your device, then firmly plug your charging cable into your device. If the charging port is damaged, your device probably needs service.
Let your device charge for a half hour (If your device is unresponsive afterward, learn what to do).
Force restart your device: Press and hold the Sleep/Wake and Home buttons at the same time for at least 10 seconds, until you see the Apple logo. Then let your device charge for another half hour.
If your device still won’t power on or charge, take your device, charging cable, and charging adapter to an Apple Retail Store or Apple Authorized Service Provider for evaluation. You can also contact Apple Support.


Source: Apple