#or whether Shakespeare cared much for a logical chronology in his plays

[noshitshakespeare]

How much time passes from the beginning of Hamlet to the end, do you suppose?

I would say at least 5 months, at most about 9 months. 

There are three sections to the play. The first, from the beginning of the play to Act 2, scene 1, appears to take place across two days. It’s just under two months after Hamlet’s father has died: “but two months dead – nay not so much, not two” (1.2.138), Claudius is upbraiding Hamlet for mourning too long because he’s mourning after the official ‘Trental’ period, which was the month fixed for the mourning of close relatives; and it appears that Claudius and Gertrude have married before this official mourning period is up: ‘within a month… married my uncle’ (1.2.145-51). 

Since Marcellus says 

Some say that ever ‘gainst that season comesWherein our saviour’s birth is celebratedThe bird of dawning singeth all night long;And then, they say, no spirit can walk abroad,The nights are wholesome; then no planets strike,No fairy takes, nor witch hath power to charm,So hallowed and so gracious is the time. (1.1.156-63)  

The action can’t be taking place during Advent. But it is very cold at night, and spirits are appearing, which suggests that the ghost is walking around All Hallow’s Eve and All Soul’s Day. The first night, Bernardo, Marcellus and Horatio see the Ghost, later that day Hamlet is in court with Claudius and his mother, and that night around midnight, Hamlet encounters the ghost and decides to put on an antic disposition. The next day Laertes leaves for England and Polonius tells Ophelia to refuse Hamlet’s attentions.

There is a gap here, and the second sequence, from Act 2, scene 2, to Act 4, scene 4, appears to take three days. Rosencrantz and Guildenstern have been sent for, and Hamlet has appeared all dishevelled in Ophelia’s closet just before the action resumes. In the mousetrap scene Ophelia says that Hamlet’s father has been dead ‘twice two months’ (3.2.121), so it’s been four months since the death of the old King, and since it was one to two months since the death of the king in Act 1, this second section happens at least two months from the beginning of the play, at most just under three months. It’s probably January, and it’s possible the players are coming in time to perform for Twelfth Night or Epiphany.

The first day, Ophelia runs to her father, Rosencrantz and Guildenstern arrive, the players arrive, and Hamlet says ‘We’ll hear a play tomorrow ... We’ll ha’t tomorrow night’ (2.2.472-476). The nunnery scene happens most likely the day after, and the play is put on that night, after which Gertrude summons Hamlet to her closet. It’s just past midnight: ‘Now is the very witching time of night’ (3.2.378). Since the ghost appears again, presumably the Christmas celebration period ended at midnight, suggesting the night the Murder of Gonzago was performed was Epiphany. Hamlet kills Polonius and Claudius says ‘the sun no sooner shall the mountains touch / But we will ship him hence’ (4.1.29-30) and does just that in Act 4, scene 3. Hamlet encounters Fortinbras’ army on his way to the ship and that very night, on the ship, exchanges Rosencrantz and Guildenstern's commission, because the next say the pirates attack (Hamlet says it was ‘ere we were two days old at sea’ [4.6.15]). 

It’s not clear how much time passes between this and the last section, which begins at Act 4, scene 5, when Ophelia comes in mad. In this sequence, Laertes bursts in, there’s the scene with Ophelia’s flowers, and Claudius receives a letter from Hamlet: ‘Tomorrow shall I beg leave to see your kingly eyes’ (4.7.44-45). Ophelia dies, and the next day she’s buried (the gravedigger scene), after which there’s the final duel. This seems to all take place in two days or so. It appears that it’s been a little while since the last section, since the King asks ‘How long hath she been thus?’ (4.5.66) when he sees Ophelia mad. Since Claudius claims that ‘Two months since’ (4.7.80) Lamord has praised Laertes’ swordsmanship ‘in Hamlet’s hearing’ (4.7.70), Hamlet can’t have been gone more than two months. At shortest he could have been back in less than a week, since he was taken by pirates no more than two days journey away. But Fortinbras has had enough time to march to Poland and back, and there’s no dating quite how long Hamlet spends with the pirates. The only reference to a particular date for this sequence is Ophelia’s ‘Tomorrow is Saint Valentines day’ song (4.5.48), which could be thematic rather than accurate dating, since she’s mad. Still, it does hold with the general season of the play, since the mousetrap sequence was probably in January. The only thing that might throw this off is the other piece of unreliable evidence, which is that Ophelia’s flowers are all summer flowers: fennel, columbine, and rue; and when the Queen describes Ophelia's drowning, she says Ophelia made garlands of ‘crowflowers, nettles, daisies and long purples’ (4.7.167), again, all summer flowers. Either these are all Ophelia's imaginary flowers, or Shakespeare isn’t too concerned about the time-frame of this final section. At the very latest it places the final action in the summer, so about 9 months after the start of the play. 

So there are two months separating the first section from the second section, and an unstated amount of time passing between the second section and third section. Given circumstantial evidence it’s most likely that the action takes place across 5 to 6 months, 9 months if the evidence of the flowers are enough to convince you.