I've made a band and have a guitarist character. I've also made a drummer and I can play the world tour with the drummer (and it will continue the bands tour) but how can I permanently add that character to the band's roster? 

You can't.

You can add him in as a stand in. Just go to your band's options and there'll be a menu for stand ins. Select your character and he will always be playing with you (you can also select any premade ones you like to play in the other empty areas).

As for having his name on the actual list of band members, I don't think it's possible.