One suggestion for the client-side implementation would be to incorporate in your app the Facebook plugin
available in the Telerik Marketplace. With this approach you will incorporate a native-like experience for the user as provided by the native iOS and Android SDKs of Facebook. The recommended way in mobile apps would be the native approach using the plugin.
Subsequently, when the user is authenticated from Facebook from the mobile app, you will need to supply the provided access token from Facebook (obtained via the plugin) to your backend for further validation of the user authentication/authorization.
Please, bear in mind that the Facebook plugin it is not runnable in the Telerik Platform simulator because it uses only native, platform-specific code. It will work only when the app is deployed on a real device as an application package.
I hope that this helps.