[Eclipse]

So I've found a way of playing YouTube in the background. @Oni's method is nice but many people don't want to pay for another app for just one feature.

I recently rooted my S3 GT-i9305, and flashed CyanogenMod-11. (Note that CyanogenMod is not required for this tutorial, but root access is.) After looking around for a bit, I stumbled upon Xposed Installer.

Xposed is a framework for modules that can change the behavior of the system and apps without touching any APKs. That's great because it means that modules can work for different versions and even ROMs without any changes (as long as the original code was not changed too much). It's also easy to undo. As all changes are done in the memory, you just need to deactivate the module and reboot to get your original system back. There are many other advantages, but here is just one more: Multiple modules can do changes to the same part of the system or app.

Install it from the link above instead of from Google Play, and once done, open it and grant it root access. It may ask you to install the framework. Just install it. Then search for 'Youtube Background Playback', or something along those lines. Click on the result, and it'll take ten seconds to install the module. After that, reboot and open the official YouTube app. Boom.

Enjoy.