Unsigned APKs do only work on rooted and patched devices/environments.
These are not always provided by the mod publisher as they do only work under certain circumstances.
1.) Your device must be rooted.
2.) Your device must be patched to ignore app signatures. This can be done with the help of tools such as Luckypatcher or Xposed.
Once you fill that requirements the process is the same as with signed APKs with the difference that you can overwrite the original game/app with the MOD APK without removing it first.
1.) Download the unsigned MOD APK.
2.) Install the unsigned MOD APK.
3.) Enjoy.
Note:
For the case the unsigned APK does fail to install: Your device patch is not done correctly!
Login methods and In-App purchases on unsigned MODs function the same as in the original game. Logging in with Google works without issues. Keeping the Facebook application installed is also no problem.
You are rooted and want to know how to patch your device? Please check this for more information:
[Videotutorial] How To Login With Google+ Or Facebook App On Modded Games