How do I check an iOS device's battery health?
There are apps that you can use to assess battery health, but I recommend a Mac/PC app called iBackupBot. Tether your device (non-jailbroken works) to your computer, navigate to the lower left window pane, select the device, and click the link for ‘More Informations’. The top portion of the resulting window will look something like this:



From Check iPad iPhone battery charge cycle count without jailbreaking - iCopyBot:


  
  Connect your iPad, iPhone or iPod Touch to computer, then launch iBackupBot.
  Select your iPad, iPhone or iPod Touch in Devices window, then click More Informations in right side window
  iBackupBot will pop up a window to show many informations about your iPad, iPhone or iPod Touch, include the battery charge cycle
  count, battery campacity and more
  


The best iOS app I have used to provide this info is BatteryLogger Plus. However, it requires a jailbreak. Non-JB apps seem to provide optimal operating information, but not cycle count information. This may be a function of App Store limitations, which is why BatteryLogger Plus is a Cydia store app instead.