In the first two scenes of the play we don't see much of Hamlet, but what we do see is what gives him the nickname the "melancholy Dane." He is obviously depressed and unhappy and sarcastic--and all for good reason. He is a little too dramatic for my taste, but what happens to him is monumental and his emotions match that.
If you are referring to the first two scenes of the play, then I would say I might want to be friends with Hamlet just because he appears clever and witty, plus I feel sorry for him in having to deal with the death of his father and his mother's quick remarriage to someone he doesn't like -- at all! I would admire his pun "I am too much in the sun" (son... he hates the idea of his uncle being his new step-father). I would try to help him deal with the anger and despair he expresses in his first soliloquy.
I think you'd have to look at how he treats his friends. Horatio, Rosencrantz and Guildenstern are each friends from his younger days. When Horatio arrives, Hamlet is very glad to see him. He takes him into confidence as illustrated here. When Horatio confirms that it is indeed him, he says:
This is a prince talking. You have to remember that!
On the other hand, when R and G arrive, he is as excited to see them, but he questions their presence, eventually pulling out of them that they are lying:
No such matter: I will not sort you with the restROSENCRANTZ HAMLET
of my servants, for, to speak to you like an honest
man, I am most dreadfully attended. But, in the
beaten way of friendship, what make you at Elsinore?
Beggar that I am, I am even poor in thanks; but IGUILDENSTERN HAMLET
thank you: and sure, dear friends, my thanks are
too dear a halfpenny. Were you not sent for? Is it
your own inclining? Is it a free visitation? Come,
deal justly with me: come, come; nay, speak.
Why, any thing, but to the purpose. You were sentROSENCRANTZ HAMLET
for; and there is a kind of confession in your looks
which your modesties have not craft enough to colour:
I know the good king and queen have sent for you.
That you must teach me. But let me conjure you, by
the rights of our fellowship, by the consonancy of
our youth, by the obligation of our ever-preserved
love, and by what more dear a better proposer could
charge you withal, be even and direct with me,
whether you were sent for, or no?
In this sense, I would say that I would want Hamlet as my friend, because he is very smart and can read people. If he were hanging out with me, I'd know I was in his good graces. And when you're talking about a prince, that is a good thing!