It’s usually best to copy the existing one, name it something unique, and then add your own functionality.
Copy the WP_Widget_Recent_Posts class from wp-includes/class-wp-widget-recent-posts.php into your functions.php (or, preferably, another file in your theme devoted just to widgets) and rename to something else, like My_Widget_Recent_Posts
Add your functionality in there.
Don’t forget to call register_widget('My_Widget_Recent_Posts') as part of the init phase as well.