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You can load the video on youtube then once you have exited youtube and gone onto another app if you double tap the home button so that the multitask thing comes up then slide it to the right and press the play button it will work but i think you have to go straight from youtube to the app you want to use otherwise it uses the music you have saved on your ipad instead of the sound from the youtube video. Hope this helps if your not too sure what i mean let me know and ill try to explain better.
I don't think there's a way to play a playlist in the background, Kay. Unfortunately, it also appears that using the new YouTube app means no more background YouTube listening. Now, you have to run the website version in a tab on Safari.