Complicated and risky plot (he teach well his future enemies). Sacking Dumbledore doesn't seem worthy such big risk. Quirrell now has the possibility to teach future wizards what he thinks will serve him, is impossible to fire, has influence on Harry, and Dumbledore doesn't seem to get much value out of being head of the Hogward.

I think Quirrell wants to unite wizards using some external enemy, and conquer the world/stop muggles from nuking themselves. Harry has great PR, so he'll be the fuhrer of united magical Brittain, and Quirrell will provide common enemy somehow to make it easier for Harry to rule.