Well, it depends really, on the audience, and on the production. Shakespeare as ever presents two sides to the Prince, even though he's only a relatively minor character. And it's worth remembering too that most of the audience would know already from their knowledge of history that the princes in the tower were going to end up dead.
So the scene takes on, depending on how it's played, a sort of ominous black comedy - either that, or just ominousness!
The prince initially comes across as a bit of a spoilt brat:
No, uncle; but our crosses on the way
Have made it tedious, wearisome, and heavy:
I want more uncles here to welcome me.
Yet he does even manage to hold his own against Richard's suggestion that the uncles he wants to welcome him were "false friends":
God keep me from false friends! but they were none.
Not many people stand up to Richard in the play. Do we admire this, or does it merely make us shiver a bit, knowing that Richard will have his revenge? Do we laugh or tremble at Richard's asides:
So wise so young, they say, do never live long.
Try reading the scene out, and you'll see what I mean. First play Richard like Mr. Punch, all jokes and smiles. Then play him totally serious and scary (like a real psychopathic murderer). The scene works both ways.