You will first need to integrate your game with the Kongregate APIs, so that Kongregate users can sign in to your game. Please refer to these resources if you need help with this:
Control Panel Setup
When you setup your Connection to use the Kongregate authentication type, you need to provide the Kongregate API key, which you can find on the API Information page for your game.
To authenticate a Kongregate user, you need to get the user id and game auth token.
Once you have those, you can call Authenticate like this: