I've searched for a good solution myself when I had this problem, and after solving it I've gotten a lot of messages of people asking how I did it, so instead of me replying the solution to everyone, I'm going to make this as basic as possible. :)
Preparation
First of all, install Cranberry's google play games plugin to Construct 2. Here's a useful tutorials for doing this:
- How to install plugins
Now I'm assuming you have a Google Play Developer license. If not, you can purchase is for 25$ (one time payment) here.
Configuring the plugin in Construct
Now you want to set your Google Play Game Services APP ID to your plugin setting in construct 2. First check your projects list and find the plugin you've added and click it. On the left bar (default to left), you will find a text-box named: Android google play app id and in there you want to paste the app id found in your Developer Console -> Game Services and press your game. Under your game-name, you'll find a 12-digit ID, copy that and paste into the C2 text-box as shown in the picture below:
Now export it via cordova. You may want to uncheck 'Minify Script' as it makes some of Cranberry's plugins defect.
Installing the Intel XDK plugin
Now let's install the plugin to your Intel XDK project.
Open Intel XDK -> Open project (assuming you have one already) -> Plugin Management -> Add new plugin -> Third-party plugins and in the Plugin-ID field, you want to type in cordova-plugin-game. When Intel XDK prompts you to add the APP ID, just paste your Google Play Game Services app ID in there aswell.
You're doing great! We're so close to being done!
Let's move on to the last step in this process!