If Shakespeare was alive today, he would write his screenplays in contemporary English and use the same framework that contemporary screenplays are written.
His themes wouldn't change, IMO. "Julius Caesar" would be either a historical movie, or one about the assassination of a president because he was acting against the interests of "the guardians of the country" (left or right) or he was too well beloved or too grandiose. "Macbeth" is another play about ambition. He is not the ruler, but he will do murder to become boss. The history plays are also about the effects of ambition, and of bad decisions made by a county's rulers. Romeo and Juliet? Two teens whose parents don't want them together.
Shakespeare knew how to write. He could captivate audiences with well told tales, and he could convey his messages though his writing. He'd have no trouble as a writer/director today.