Who does Hamlet have a false friendship with other than Rosencrantz and Guildenstern.?

Hamlet does not have false friendships with anyone. Lots of people are false with him, but he is never false with them.

Rosencrantz and Guildenstern are Hamlet's school friends. King Claudius has sent for them so that they can spy on Hamlet and Hamlet knows it. In Act 3, scene 2, he says to them:

HAMLET:

Why, look you now, how unworthy a thing you make of me! You would play upon me; you would seem to know my stops; you would pluck out the heart of my mystery; you would sound me from my lowest note to the top of my compass; and there is much music, excellent voice, in this little organ, yet cannot you make it speak. 'Sblood, do you think I am easier to be played on than a pipe? Call me what instrument you will, though you can fret me, you cannot play upon me.

They are the ones who are false, not Hamlet; they are the ones feigning friendship, not Hamlet.

Falsehood is not one of Hamlet's traits. Indeed, he is, in his own words, "too much in the sun." If anything he is too honest. In Act 2, scene 2, he laments to Polonius:

HAMLET:

Ay, sir. To be honest, as this world goes, is to be one man picked out of ten thousand.

He sees himself as an honest man, a rare being in the world he inhabits. He is surrounded by spies and liars. He has trouble trusting anyone. In Act 3, scene 1, when he chastises Ophelia for being untrue and for being used by others, he says of himself:

..I am myself indifferent honest, but yet I could accuse me of such things that it were better my mother had not borne me...

Yes, honesty is a highly respected quality for Hamlet, and he does not deal falsely with others. Quite the contrary.