How did william shakespeare show renaissance influences in his work?

Shakespeare's plays are all about questioning authority: kings are deposed; bad people (Iago) triump over good ones (Cassio); your parents don't always know best (the behaviour of the parents in Romeo and Juliet is the cause of all the trouble).
In the Middle Ages people had a general sense that God was in his heaven, and all was right with the world. In the Renaissance people started to ask if that was true.
Shakespeare is always asking difficult questions, which is a very Renaissance thing to do. And he never makes any direct reference to Christian faith in any of his plays:- religious doubt was also a very Renaissance characteristic.