I am playing in a fictional league and assigned myself to a team. I am considered the GM. When playing out games, I can assign strategies, etc. But I am not listed as the Head Coach. Is there a way that I can be the Head Coach as well? What do I need to do? Can this ever be done at the start of a new league?

You need to fire your coach, then hire yourself as coach on the Staff screen.

I fired the coach, but was not able to hire myself. I do not show up anywhere to be hired. The button "hire GM as coach" stays grey.

I see that as well now when I tried it. I've contacted Gary.

Are you playing in Global mode or Local mode? In Global mode you can't assign yourself as the head coach but you have all the functionality of the HC. You can just choose on the strategy screen to use the hired HC's strategies or your own but you can change the lineups, coach the games...everything the HC can do. In Local mode you only have the role you are assigned to - in that mode if you are the GM you can fire the HC and then hire yourself as HC to take over those jobs.