To backup to iCloud, first, you will want to make sure you have the Files app downloaded onto your device. Then you can go into OnSong > Tap Share Menu > Export - tap the folder icon > Select Files > OnSong Backup > Export > Select iCloud Drive -If you have a specific folder created you can then tap that, if not... > Tap Add.
By the way, you can do all of your exporting this way... not just for backups.
In OnSong 2020: You can also import from iCloud as well, either directly from the app itself or if you go into OnSong > Tap Songs > All Songs > "+" Sign > Import/Files > Select content > Import.
In the latest version of OnSong: You can also import from iCloud as well, either directly from the app itself or if you go into OnSong > Tap Songs > All Songs > "Box/Arrow" Sign > Import/Files > Select content > Import.