There is no 100% effective remedy as the source of the problem can be very uneven but we must proceed as follows to detect the problem:
Clear data and cache of Google Play Services - We are going to Settings -> Applications (or Application Manager) and select the All tab or All to search Google Play services as we can see in the picture. There seek Clear Cache option and the option Erase data can sometimes be Managing storage that will lead to the choice of settings where Google itself is the option to erase all data from Google Play services.
Clear data and cache Play Store - Once done repeat this step but looking for the Play Store application rather than search services Google Play.
In most cases this solves the error. Restart your phone and confirms that the application does not stop, otherwise continue on the following list:
Repeat step 1 and 2 and uninstall updates both updates. Button to uninstall the updates are in the same place as that of Clear cache and Clear data even top. Once this is done, and finally, Clear data and cache Google Services Framework Play (another application that you can find in the same place as before). Restart your smartphone and your phone should ask an update of Google Play Services you must install and in this case should not fail again.
Tell us if the tutorial has worked for you, otherwise you may have to ask for help in the forums of Google but it is very unlikely that you did not work out. Just make sure that your Internet connection is stable and has enough speed or otherwise could give errors when upgrading or download large applications. The tutorial has been done on a Samsung Galaxy S6 but the options in Android native or other layers of customization are similar.