TL;DR: Aristotle’s Poetics was available in translation, but there is no evidence that Shakespeare read it. However, it is likely that he was familiar with some of Aristotle’s ideas via secondary sources like Philip Sidney’s The Defence of Poesie.
Possible misconceptions in the question
The question can be read as implying that observing (or not) the ‘five-act division’ of plays, and the ‘three unities’, would be evidence for having read Aristotle’s Poetics (or not). But this isn’t the case, because these dramatic rules do not come from Aristotle!
The rule that a play should have five acts comes from Horace:
Neue minor neu sit quinto productior actu
fabula, quae posci uolt et spectanda reponi;
[Let a play which would be inquired after, and though seen, represented anew, be neither shorter nor longer than the fifth act.]
Horace, Ars Poetica ll. 189–90. Translated by C. Smart & Theodore Alois Buckley (1863).
Of the ‘three unities’ (action, time and place) only the unity of action is expressed in the form of a rule in Aristotle:
After these definitions we must next discuss the proper arrangement of the incidents since this is the first and most important thing in tragedy. We have laid it down that tragedy is a representation of an action that is whole and complete and of a certain magnitude, since a thing may be a whole and yet have no magnitude. A whole is what has a beginning and middle and end. A beginning is that which is not a necessary consequent of anything else but after which something else exists or happens as a natural result. An end on the contrary is that which is inevitably or, as a rule, the natural result of something else but from which nothing else follows; a middle follows something else and something follows from it. Well constructed plots must not therefore begin and end at random, but must embody the formulae we have stated.
Aristotle, Poetics 1450b. Translated by W. H. Fyfe (1932).
The other unities were developed by the dramatists of Italian Renaissance and written up as elucidations of Aristotle. They were put into the form of rules by Lodovico Castelvetro in his commentary on the Poetics:
la tragedia, la quale conviene avere per soggetto un’azzione avenuta in picciolo spazio di luogo e in picciolo spazio di tempo, cioè in quel luogo e in quel tempo dove e quando i rappresentatori dimorano occupati in operazione, e non altrove né in altro tempo.
[tragedy ought to have for subject an action which happened in a very limited extent of place and in a very limited extent of time, that is, in that place and in that time, in which and for which the actors respresenting the action remain occupied in acting; and in no other place and in no other time.]
Lodovico Castelvetro (1570), Poetica d’Aristotele vulgarizzata e sposta. Translated by H. B. Chalton (1913). Castelvetro’s Theory of Poetry, p. 85.
Could Shakespeare have read Aristotle’s Poetics?
In French or English translation? No. The first French translation was La poétique d'Aristote (1671) by ‘le sieur de Norville’, and the first English translation was the anonymous Aristotle’s Art of Poetry (1705), “Translated from the Original Greek, according to Mr. Theodore Goulston’s Edition, Together With Mr. D’Acier’s Notes Translated from the French.”
In Latin? Yes. There is strong evidence that Shakespeare was able to read Latin:
we cannot question the competence of Shakespeare’s Latin, small as it may have been by Ben Jonson’s prodigious standards. Latin was the substance of the grammar-school curriculum; and within that curriculum, Ovid occupied a very special place […]. Shakespeare got a good enough education for him to be able to base his Lucrece on a story in Ovid’s Fasti, which was not published in an English translation until well after his death.
Jonathan Bate (1993). Shakespeare and Ovid, p. 13. Oxford: Clarendon.
There were Latin translations of the Poetics, for example Aristotelis Poetica by Alessandro de’ Pazzi, published 1536. That these translations were of interest to Shakespeare’s contemporaries is demonstated by the fact that Ben Jonson owned
A Latin translation of the Poetics by the English scholar Theodore Goulston
David McPherson (1974). ‘Ben Jonson’s Library and Marginalia: An Annotated Catalogue’. Studies in Philology 71:5, p. 26.
This was a parallel Latin–Greek edition, the one used as a basis for the anonymous 1705 English translation, but as it was published in 1623 it arrived too late for Jonson to have lent his copy to Shakespeare.
In the original Greek? Probably not. In his epitaph ‘To the Memory of My Beloved the Author, Mr. William Shakespeare’, Ben Jonson described Shakespeare as having “small Latin and less Greek”, and the scholarly consensus is that
the possibility that Shakespeare read a Greek text of the Poetics is extremely remote. His formal studies extended only as far as grammar school in Stratford, and since he did not attend university, we have no clear evidence that he knew Greek. In what remains the most authoritative and comprehensive study of Shakespeare and the classics to date, T. W. Baldwin argues that Shakespeare probably studied some Greek at school. However, Baldwin acknowledges that this is likely to have been New Testament rather than classical Greek, and this would not have equipped Shakespeare to read Aristotle, or indeed, the texts of Greek drama, in the original.
Sarah Dewar-Watson (2004). ‘Shakespeare and Aristotle’. Literature Compass 1, p. 2.
In Italian translation? Perhaps. There is some evidence that he could read Italian:
The best evidence that Shakespeare could read Italian, however, comes from the close adherence of his plays to his Italian sources. For some plays, those Italian sources had not been translated into any other language, and the only logical conclusion is that Shakespeare must have read the source in Italian. In other instances, although the Italian source had been translated into French or English, Shakespeare’s play is often closer to the Italian original than to the translations or adaptations of the original.
Naseeb Shaheen (1994). ‘Shakespeare’s Knowledge of Italian’. Shakespeare Survey 47, p. 163–4. Cambridge University Press.
If this is right, then he could have read the Poetics in the 1570 Italian translation of Lodovico Castelvetro.
Did Shakespeare read Aristotle’s Poetics?
It is hard to prove a negative. All we can say is that no-one has identified any passages in Shakespeare that are clearly derivative of Aristotle:
it is also widely agreed that Shakespeare never read the Poetics: there is no linguistic evidence to suggest that he did.
Dewar-Watson, p. 2.
What about secondary sources?
Even if Shakespeare never read the Poetics, that does not mean that he was unaware of Aristotelian and neo-Aristotelian ideas:
it was long a favourite idea that Shakespeare was totally ignorant of the ‘rules.’ Yet this is quite incredible. The rules referred to, such as they were, were not buried in Aristotle’s Greek nor even hidden away in Italian treatises. He could find pretty well all of them in a book so current and famous as Sidney’s Defence of Poetry [sic]. Even if we suppose that he refused to open this book (which is most unlikely), how could he possibly remain ignorant of the rules in a society of actors and dramatists and amateurs who must have been incessantly talking about plays and play-writing, and some of whom were ardent champions of the rules and full of contempt for the lawlessness of the popular drama?
A. C. Bradley (1904). Shakesperian Tragedy. London: Macmillan.
Sidney elucidates concepts from Aristotle’s Poetics, and mentions the unities too:
Our tragedies and comedies not without cause cried out against, observing rules neither of honest civility nor of skilful poetry, excepting Gorboduc,—again I say of those that I have seen. Which notwithstanding as it is full of stately speeches and well-sounding phrases, climbing to the height of Seneca’s style, and as full of notable morality, which it doth most delightfully teach, and so obtain the very end of poesy; yet in truth it is very defectious in the circumstances, which grieveth me, because it might not remain as an exact model of all tragedies. For it is faulty both in place and time, the two necessary companions of all corporal actions. For where the stage should always represent but one place, and the uttermost time presupposed in it should be, both by Aristotle’s precept and common reason, but one day; there is both many days and many places inartificially imagined.
Philip Sidney (c. 1579). The Defence of Poesie.
There are passages in Shakespeare that can be interpreted as apologies for rule-breaking, which implies that he was aware of the rules:
In Henry V, the Chorus explicitly acknowledges the play’s breach of the Unity of Place: ‘Linger your patience on and well digest/Th’abuse of distance, and we’ll force our play’ (2.0.31–32). This reference suggests that Shakespeare’s failure to observe the Unities is part of a deliberate strategy, rather than a sign of his theoretical ignorance. This is echoed in The Winter’s Tale, in which the figure of Time makes explicit reference to the play’s breach of the Unities (4.1.4–8):
Impute it not a crime
To me or my swift passage that I slide
O’er sixteen years, and leave the growth untried
Of that wide gap, since it is in my power
To o’erthrow law
Dewar-Watson, p. 6.
