Depends on how your old site was built. If it's made to modern web standards with HTML5 and CSS layouts, you might be able to drop your content into a new CSS layout. If not, then you're looking at a complete re-build.
Make your existing site Responsive
http://blogs.adobe.com/dreamweaver/2015/08/making-an-existing-site-responsive-with-new-capabilities-in-dreamweaver-cc.html
But keep in mind that a responsive layout doesn't make your site mobile friendly unless forms, navigation menus & content are scaled appropriately for smaller touch screen devices. A lot of careful planning & streamlining goes into making a good responsive web site. Mobile users interact differently with their devices than desktop users. They require less content, bigger links and images/media that won't drain their data plans.
My advice is to read all you can online about RWD (responsive web design) and look at what other web developers are doing to see what works best.
Responsive Web Design Guidelines and Tutorials – Smashing Magazine
W3 Schools - Bootstrap
http://www.w3schools.com/bootstrap/
Tutorial Republic - Bootstrap for Beginners
http://www.tutorialrepublic.com/twitter-bootstrap-tutorial/
Nancy O.