Will Prince William succeed Queen Elizabeth II?

There's a chance he might do so directly. Prince William is second in line to the throne, after his father, Prince Charles. When Queen Elizabeth II passes on the crown, by death or voluntary act, one of the following will happen.

1 - Prince Charles takes the throne as King Charles III. Prince William is therefore the heir apparant, the next in line. Barring his own death or other exceptional circumstances, he will eventually become King.

2 - Prince Charles refuses the crown, allowing William to directly succeed from Queen Elizabeth II.