Writing your own home screen application is possible. It is called the Launcher.
You can get the source code of the default Android Launcher via Git.
The project URL is:
https://android.googlesource.com/platform/packages/apps/Launcher2.git
You can get the source code like this:
git clone https://android.googlesource.com/platform/packages/apps/Launcher2.git
That will create a directory called Launcher2 for you. Now you can get cracking and create your custom launcher.
If you need help with using Git then checkout Git’s documentation section.