Polonius is a talker. (Probably a close talker, too.) He dishes out cliché advice to his son, Laertes, before the kid heads off to college ("to thine own self be true," "neither a borrower nor a lender be," and so forth), and he often speaks in an overly ornate way when telling stories. Even the ever-gracious Queen Gertrude has to cut him off at one point by saying "More matter, with less art" —code for "cut it out and get to the point."
What does this say about Polonius? He's a bit of a buffoon and a lot of a poser. He wants to be taken seriously but he actually just sounds like a foolish old dolt. And he doesn't stand a chance in the highly competitive and political world of the court. (Eventually, he gets himself stabbed in the gut when he tries to spy on Hamlet.) King Claudius, on the other hand, is a guy who knows how to talk. He delivers well-wrought speeches that are in keeping with his deceptive nature:
Though yet of Hamlet our dear brother's death
The memory be green, and that it us befitted
To bear our hearts in grief and our whole kingdom
To be contracted in one brow of woe,
Yet so far hath discretion fought with nature
That we with wisest sorrow think on him,
Together with remembrance of ourselves. (1.2.1)
The speech starts out by acknowledging Old Hamlet's death and the grief felt by the entire kingdom, but check out the way Claudius manages to sweep Old Hamlet under the rug when he says it's time to move on from all the boo-hooing.
Basically Claudius is saying something like this: "It's fine to think on Old Hamlet with sorrow, but, really, let's think about ourselves." Yeah. We almost didn't catch that last part either. Claudius is pretty crafty, which is why he's got everyone (except Hamlet) eating out of the palm of his hands. It's no wonder Claudius has talked himself into the throne and into Gertrude's bed.
Sometimes, actions are pretty straightforward. King Claudius, for example, is easy to figure out. He kills his brother and helps himself to his dead bro's crown and wife. Tsk, tsk.
Hamlet's actions, on the other hand, are super vague. Why does Hamlet pretend to be a madman? Why does he abuse Ophelia? Why does he delay killing Claudius? We don't know for sure. What we do know is that while Hamlet is not killing Claudius, he spends a whole lot of time thinking, which alerts us to the fact that he's introspective, philosophical, and deliberate. In contrast, Fortinbras and Laertes are swift to take steps toward avenging their fathers' murders. Are we supposed to be glad that Fortinbras is going to take the crown? Or is there something to be said for Hamlet's more contemplative style?