The Google Play store has several apps that support background playback thus letting you use YouTube as a music player.
The uListen app has a simple interface and minimal features Just use the search button to find videos and playlists on YouTube and hit the play button. You can also tap and hold a search result to favorite it or to add it to your playlist. uListen doesn’t link to your YouTube account and all your playlists and favorites are saved locally.
Then there are apps like SuperTube and Viral that more like clones of the official YouTube app except that they also let you watch or play YouTube videos while in the background. You can create local playlists or connect these apps to your main YouTube account and access your playlists, watch later queue and favorites. Another handy feature is that you can do a vertical swipe on the video to increase or decrease the volume.
A major downside of the previously mentioned uListen app is that it doesn’t offer autocomplete mode so you have to the type the entire search terms manually for relevant results. YT Player, another free YouTube audio player, fixes that problem. It pulls in search suggestions from YouTube and you can search for videos, playlists or even channels. If you prefer apps with slightly bigger fonts, you will probably like this one even more.