Why is Don Pedro so melancholy?
The straight answer is 'we just don't know'. It's up to the actor and the director in any given production make a decision about him, and play that. But whatever is making him sad is in the subtext. There are two main clues, I For a start, he seems to be "friendly" towards Claudio in an incredibly passionate and committed sort of way:
My love is thine to teach. Teach it but how,
And thou shalt see how apt it is to learn
Any hard lesson that may do thee good.
It's difficult not to notice too the speed with which he joins in with Claudio to decide that Hero is a whore.
He also makes a rather sad, immediately rejected marriage proposal to Beatrice at the party:
I would rather have one of your father's getting.
Hath your Grace ne'er a brother like you? Your father got excellent husbands, if a maid could come by them.
Will you have me, lady?
No, my lord, unless I might have another for working days: your Grace is too costly to wear every day. But I beseech your Grace pardon me. I was born to speak all mirth and no matter.
But that isn't the only suspicious thing: at the end, after Claudio is remarried to the newly-alive Hero, everyone else is happy and the ending has worked out. And at the end, he's left with no wife at all. Benedick shouts to him, just before the end,
Prince, thou art sad. Get thee a wife, get thee a wife!
And he doesn't respond. Maybe it's because Don John, his brother, has just been recaptured and will have to be punished. Maybe it's because he was in love with Claudio or Beatrice; and now both of them are married. Who knows? Make of it what you will!
Hope it helps!