- Open the video you want on safari
- press the home button to exit safari, the music will stop
- swipe up so the menu thing with the brightness control etc. comes up
- press play
If you want to browse the Internet while you do it just use the chrome app or something.
They're free on the AppStore and there have never been ads and the music has always kept playing in the background. Until 2015 you could also download videos at full quality but they removed that feature. You can download videos from YT anyway using the internet. I've never used the YouTube App in my life.
The service will also have exclusive videos from YouTube stars like PewDiePie, Lilly Singh and the Fine Brothers that aren’t available on the free version of YouTube.