First and foremost, the premier portrayal of the prince's previously unprecedented preoccupation with his precariously propagating and particularly pressing predicament (possibly his primary priority) is probably represented at the point of his initial speech. He calls his uncle 'less than kind' (I, ii, 65), most likely out of a swelling pain inside him that the king would allow Laertes to go off to France and do his business, but he would not give Hamlet permission to go. To make matters worse, there is no place in the castle for Hamlet to go either. To use a royal bush would be disrespectful, and would breach Hamlet's high moral standards.
Then, after Hamlet and the king and queen talk for awhile, everybody gets up and leaves, leaving Hamlet all alone. Hamlet then proceeds to speak to himself, mainly because there is nobody left in the room to talk with. His speech to no one in particular is further indication that Hamlet requires relief. He is under so much pressure that he has begun to talk to himself. In his speech, he exclaims 'O God!' (I, ii, 132), which is clearly a desperate plea from Hamlet to God, asking for divine intervention and a place to go. Hamlet goes on to say 'Ere the salt of most unrighteous tears had left the flushing of her galled eyes.' (I, ii, 154-155). The majority of that sentence is irrelevant and can be ignored. The key word is 'flushing'. From his use of the word 'flushing', there can be no doubt as to what is on Hamlet's mind.
However, you may still have doubts as to what is on Hamlet's mind, let me further try to persuade you. Later on in the play, Hamlet speaks to himself yet again. According to the text, Hamlet says to himself 'To be or not to be, that is the question;' (III, i, 56). However, I consider this to be a mistranslation. I believe that the correct translation from whatever language this play was originally written in should be: 'To pee or not to pee, that is the question;'. It would make so much more sense, and would follow with all that aforementioned stuff that I just said previously. Hamlet goes on to say 'a sea of troubles' (same speech, a little further down). His use of water imagery to describe his situation can be nothing other than an attempt to express his frustration.
Later, Polonius (the king's number one guy), comes up with a brilliant scheme to hide behind a curtain and spy on Hamlet talking with his mother. While he is spying on them, some stuff happens, he cries out 'ho!' (III, iv, 22 or 23) Hamlet mistakes this for 'hhhooooooooo' (expression of relief) and kills him. Many argue that Hamlet killed Polonius because he thought he was Claudius, and that it was a case of mistaken identity. I tend to disagree. The truth is that Hamlet thought that whoever was behind the curtain was making use of the facilities. This thought inspired such an intense jealous reaction in Hamlet that he would have killed that person no matter who they were.
Later, Hamlet talks to some unnamed captain guy. Then the captain guy leaves and Hamlet is alone again, and he speaks to himself, again. The first thing he says is 'How all occasions do inform against me' (IV, iv, 30something), implying that even fate is against him in his quest by denying him access to a bathroom.
In the final scene of the play, Hamlet says, 'If it be now, 'tis not to come; if it be not to come, it will be now;' (V, ii, 210-211), which can be interpreted simply as Hamlet's way of saying 'Man, I really gotta pee.' He also comes right out and tells Horatio that he desperately needs to go to the bathroom when he says 'I'm going to explode if I don't find an outhouse soon' (somewhere near the back).
By now you should be convinced without a doubt that Hamlet's suffering is obviously due to his need to relieve himself rather than a series of trivial catastrophes. If not, then you probably have some doubts as to the validity of my argument. I assure you, these doubts are well justified. But enough of my pointless rantings, I must go rid myself of about a gallon of iced tea.