Inspired by Tridion Stack Exchange Question.
Registering Tab Initialize Extender is an easy process.
The only actions you should take are to implement your extensibility functionality, put it into the class with the initialize method and register this class as a Tab initialize extension.
When a page (or any other instance of Tridion.Controls.DeckPage) gets initiated, it will check for these object initialize extensions. If there are any, it will initialize them, by calling initialize method of this extension.
To register this extension just call the static method registerInitializeExtender of Tridion.Controls.DeckPage class, with a target tab identifier and your extension class name.
Then just make sure that your extension file is loaded within the functionality you want to extend.
As an example, you can check the source of Extended Area from the previous tutorial.