What purpose does the subplot of the relationship between Polonius and his children serve?
One of the interesting ideas explored in the play is family relationships. Hamlet and his relationship with his family is broken. The relationship of Polonius and his children, on the other hand, seems like a good one.
Does he love his children? Yes. Does he know how to love them? Maybe not.
On the surface, Polonius would appear to be a doddering old fool but the advice he gives is very good advice. He is like any father sending his young son away to school. Does he trust his son? Of course not since most men remember their own youthful indiscretions But most fathers don't send out a spy.
His relationship with his daughter is different. Like any father, no young man is good enough and that includes Hamlet. How long he has been a single parent isn't known but being a single father to a young female could not be easy.
A more telling bit of information is how he addresses his children. Does he use the familiar thee or thou or the more generalized you?
As for the importance of the subplot, without these three characters, the play would be missing a love interest for Hamlet, a foil for Hamlet and a father who wants only the best for his children as opposed to a step father who wants Hamlet dead.
Polonius' relationship with Laertes and Ophelia is a subplot which heightens the tones of dishonesty and distrust in the play, and speaks directly to a theme of corrupted authority.
First of all, after a lengthy speech given to Laertes about morals and virtues, Polonius then sends a servant to go spy on his son while in school. He even plans out the lie this servant should tell in order to fool Laertes. Later, Polonius hides behind a curtain to spy on his love-sick daughter and Hamlet.
While it is a little difficult to take Polonius' character seriously, the hypocrisy and humor that he provides to the story is no doubt intentional. There is dramatic irony in the fact that Shakespeare builds him up to be disliked because of (among other things) the way he treats his children, and then has his son fighting at the end to avenge his death. Polonius' lack of trust of his children is likely driven by his own propensity to lie. He is a minor character who contributes to the bigger story in a subtle but very specific way.
fortinbras and laertes serve as foils to hamlet. the families of fortinbras and (to some extent) laertes both continue to focus on the health of their state. whereas hamlet (and his "family") or more wrapped up in their own problems.
the foretells doom for norway. one would hope leaders would lead. hamlet is out bemoaning and claudius is out "wassailing" (getting intoxicated). not good for the danes.