Responsive Web Design (RWD) is a service that comes as standard with all the products that I develop, and the method created by Dave Green is an excelent way to make sure that your background images adhear to the same responsive rules as your embedded images.
Dave has wrote an excellent article on how to implement responsive images for background Images, with the upshot of increasing page load time.
Responsive Images on any Website
Dave’s article focuses on implementing responsive images within WordPress, but the example Code Pen Dave has created can work with any website.
Here is a fork of Dave’s original Pen:
Responsive Images with WordPress
In Dave’s (now removed) article he provided some great Gists that show how you can create functions for WordPress that will do all of the heavy lifting for you. I have made a fork of the RWD Background Image Gists here (because I fully intend tinkering with the code).