If you're not interested in installing a new app than youtube, just go away then, because we need a new app. - Go to appstore
Go to search
Search for "jasmine"
Click on your new app "jasmine"
Search for a song
I'm finding a Rascal Flatts song
The song "The broken road"
It is playing now
Leave the app, and double click on your home button so the multitaske bar comes up, then scroll to the left till you find the Jasmine app and pause button, and then press play
Now it is playing then press the home button again.
Lock your phone and it is still playing. Double click on the home button to pause it.
Hope the tip helped you a bit. If it is possible on the Youtube app in app store,, my mistake then. My iPhone language is in Danish, if it confuses you I'll change the pictures. Twitter: @neklasd
UPDATE! iTube is a better choice, since Jasmine doesn't properly work in iOS 6.1