How do I play a YouTube video in the background on my iPad?
Try the following steps :

1: Start the video from YouTube whilst in Safari.

2: Hit Home key once. You'll exit Safari and playback will stop.

3: Double-tap Home key to bring up the app-switcher. Slide to the right to bring up background audio controls. The icon on the right side will show whichever program you last used with background audio: Safari.

4: Press Play. Youtube audio will resume playback.