Aristotle's education of Alexander is recounted principally by two ancient authors (whose work survives): Plutarch and Diogenes Laertios. If you want to really get into a load of epistemological weeds and a bunch of philosophical musing about what it means to 'know' anything, then sure, we don't know Aristotle tutored Alexander, but by conventional methodology the weight of evidence is pretty clear that Aristotle was Alexander's tutor and they shared a reasonably close bond.
Aristotle met Alexander because he had been hired as his tutor by Alexander's father, Philip II, in 343/2 BCE. This is where Plutarch starts things as regards Aristotle in the Life of Alexander, but Diogenes Laertios gives us a little more backstory. His short biography of Aristotle, citing the lost Lives of the mid-3rd century biographer Hermippos of Smyrna, suggests that Aristotle had been part of an Athenian embassy to the Philip's court, and had been there in an official capacity. Whether this is necessarily true is a little unclear; what we do know is he had departed Athens around the time of his teacher Plato's death in 348/7, and he seems to have hung around Asia Minor for a time before Philip sent his invitation. However, his connections with the Macedonian court actually went back further: his family was not from Athens but rather Stageira in the Chalkidike, and his father, Nikomachos, was the court physician of Philip's father, Amyntas III. We don't know if Aristotle lived with his father at this time, but it still meant he did actually have a few connections with the royal court to draw upon.
Aristotle's tutoring of Alexander is not recounted at immense length, but both authors regard it as having been reasonably extensive. As Plutarch put it, 'the equipment that he had from Aristotle his teacher when he crossed over into Asia was more than what he had from his father Philip'. He singles out two particular areas where Aristotle's instruction was influential: the first were the 'acroamatic' teachings, a rather unclear set of esoteric teachings which, according to an alleged letter from Alexander to his teacher, were instrumental to his manner of rule; the second was an interest in medicine, and Plutarch alleges that Alexander often administered treatment to his sick and wounded friends. He also states that among Alexander's most prized possessions was an edition of the Iliad that he is supposed to have kept under his pillow, and which he later stored in a jewelled case looted from Darius III's baggage after the Battle of Issos in 333 (possibly – Plutarch is somewhat evasive about his sources here). There are, however, certain caveats to keep in mind: Plutarch, especially in the early part of the Life of Alexander, seems to point to Alexander's education by Aristotle as part of his portrayal of Alexander as a Panhellenic hero, by making him out to be a particularly devoted student of one of the most quintessentially Greek philosophers. But the image is more complicated than he makes out, as Plutarch himself would admit.
Around 327, Alexander apparently began – if he had not done so already – to have an increasingly strained relationship with Aristotle, who had returned to Athens before Alexander left for Asia, but with whom it seems he was still corresponding. The 'conspiracy of the pages' in 327 saw Hermolaos, one of Alexander's servants, attempt to assassinate him, and according to Plutarch this event was alleged to have been instigated by Kallisthenes, Aristotle's great-nephew and one of Alexander's courtiers. Kallisthenes was placed under arrest and was, in Plutarch's account, supposed to have been put on trial with Aristotle in attendance when Alexander returned to Greece, but he died of disease while the army was in India. (Diogenes Laertios adds the grisly detail that Kallisthenes may have been thrown to a lion when it became clear he would not survive the journey.) Why exactly Plutarch regards this as a sign of growing hostility is a little unclear, and indeed it doesn't really have much of a payoff.
There was a persistent rumour in the ancient world that Aristotle had actually been partly responsible for Alexander's death. Both Plutarch and Arrian include – but dismiss – a narrative in which Aristotle was supposed to have advised Antipatros, Alexander's governor holding the fort in Macedonia and Greece, to assassinate the king, to unclear ends. Much ink has been spilled over the general credibility of this claim: Aristotle's involvement is probably fictional, but the claim that there was a murder plot concocted by Antipatros – whose son Iollas was one of Alexander's cup-bearers – has been treated by some as potentially credible, in spirit if not in practice. Christopher Blackwell's In the Absence of Alexander makes a reasonably convincing argument for there having been a considerable erosion in Alexander's authority in Greece, culminating in the effective takeover of that part of the empire by Antipatros, a process rather suddenly derailed by the total fragmentation of the empire anyway in the wake of Alexander's death. While we cannot prove Antipatros did poison Alexander, it's oddly plausible that he might have done. Alternatively, this was a claim fabricated entirely by Antipatros' rivals, such as Alexander's mother Olympias, as a means of discrediting his pretensions to power, and all of the preceding means nothing at all.
What we do know, from Diogenes Laertios, is that whatever the relationship became by the end, it was, in the ancient world, a pretty well-attested one, with Aristotle's surviving oeuvre by his day (ca. 200 CE) including a petition to Alexander and four volumes of correspondence with him. Unfortunately, these are simply not quoted or alluded to at length by any of the major literary sources, so our understanding of Aristotle's influence on Alexander is pretty much limited to the following:
A popular narrative is that it was Alexander's abandonment of Panhellenism in favour of a more universalist, multicultural manner of rule that put off Aristotle, who had counselled a much more ethnocentric policy on Alexander's part. While Plutarch, in On the Fortune or Virtue of Alexander, does suggest that Alexander differed from his tutor in this regard, there simply isn't an explicit link in the sources between this political difference and their falling-out, presuming that this falling-out even took place at all.