How to change an existing theme to a responsive bootstrap theme?
If you want to create a responsive site in drupal with bootstrap.then follow steps:


Install bootstrap theme.(https://www.drupal.org/project/bootstrap). 
Copy the bootstrap_subtheme, Place in theme folder and rename with your subtheme.
3.Copy and include the css and js of the existing site to the bootstrap_subtheme.
See the structure of the existing site and create regions in the new theme according  to requirement and apply the css.
Check the section one by one and apply the css according.
Like this you can easily convert the non-responsive site to responsive.


Notes:


You will be create a regions in the tpl.php , this Link will help you.
In this theming there is no that much different to normal theming . if you install bootstrap then it automatically give you the responsive platform, only one difference you can use the bootstrap class in html structure and also use the bootstrap module. Follow This.