How do I create Widget within plugin that uses its own class?

There’s no need to nest classes (and you can’t anyway), you just create a new instance of the class. The code below would automatically call the My_Widget class to create a widget based on your existing code.

Class plugin_name {
        // Call the widget class
        public function __construct(){
             $this->widget = new My_Widget();
             $this->widget->getWidget();
        }
}

Class My_Widget extends WP_Widget(){
     public function getWidget(){
          //Do my widget stuff.
          echo "I'm a widget";
     }
}