I’m developing a Android application, and I would like that when it first starts it set itself as default action for Home button.
Currently this option is given to the user by the Android system only when he first press the Home button after running my app.

User don’t want to click on check box as “Use by default for this action” it should be set by default my apps.
Thanks in advance.
This is not possible, and it is a good thing too.
Android allows users to choose their own defaults to prevent malicious apps from making themselves the default app to deal with an action, and hence protecting the user’s data and device. If an app could set itself as the default homescreen or browser or call app etc. then it could steal the user’s personal data and also potentially spoil or deny access to the device itself.
Due to the above reasons, this is not possible why the Android SDK, and I sincerely doubt that an unofficial means to do this exists.