I am trying to create 4 buttons as shown on the image below. Each button is clickable. How can I go about this.

There is no straight way to achieve this. However you can achieve this by doing some workaround and help of layouts. You can check one of my answer with similar problem.
Here you can take ImageButton widget of Android.
And u can set any Image in background using background:src.
That is done what you want.
You can extend Button class and write your own customized button. This may help http://developer.android.com/training/custom-views/index.html