The former King (King Hamlet) is dead and his brother, King Claudius has taken the throne and married Gertrude (King Hamlet's then wife). Claudius does not believe that Prince Hamlet should be overly sad about his father's death.
Claudius has taken over King Hamlet's throne and has married Gertrude. Claudius is very compassionate towards how Prince Hamlet feels and tells the prince to take as long as he needs to recover.
Claudius is not yet king and hates King Hamlet. Gertrude is Claudius' girlfriend and Claudius doesn't believe that Prince Hamlet is seriously grieving his father's death
Gertrude is the queen and she has already decided not to announce a new king. Thus, Denmark does not have a King at the moment and Prince Hamlet is going to become King.