This is a good question. It is also one that is going to be hard to answer in a decisive, satisfying way.

Firstly, let's get the preliminaries out of the way. Diogenes and Alexander were contemporaries. They were alive at the same time -- and the legend goes that they died even on the same day.

Also, they were both active in the same part of the world. The meeting was said to have happened in Corinth, where Diogenes indeed was at this point in his life. Alexander did physically enter the Peloponnese, where, after his forces' victory at Chaeronea in 338 BC, he was broadly welcomed (by everyone except Sparta). The league he founded was the League of Corinth, so he presumably really was in Corinth at the same time that Diogenes was.

To be specific, Robin Lane Fox's biography Alexander the Great (1973) has it that the meeting took place in 336 BC, the one and only time Fox thinks Alexander was in Corinth. Note that this changes the framing of the meeting: Alexander is not a king, but he is the 20-year-old son of Philip II. Yes, he is proving himself in Greece at the time, but he's hardly the Alexander the Great yet.

We know, then, that it was certainly possible that the two met. Well, what was the substance of the meeting?

Here's where it gets trickier, and here's why I think a reasonable person would think that the details are apocryphal. There are different versions of the meetings that report different words that Diogenes said.

Diogenes Laertius' (whose name should not be taken to mean that he was Diogenes himself; very different people!) account has it that Alexander asked Diogenes what he wanted so that he could fulfill it for him. Diogenes replies: "stand out of my light" (VI 38). In Greek: "ἀποσκότησόν μου"

Plutarch's story: 'Thereupon many statesmen and philosophers came to Alexander with their congratulations, and he expected that Diogenes of Sinope also, who was tarrying in Corinth, would do likewise. But since that philosopher took not the slightest notice of Alexander, and continued to enjoy his leisure in the suburb Craneion, Alexander went in person to see him; and he found him lying in the sun. Diogenes raised himself up a little when he saw so many people coming towards him, and fixed his eyes upon Alexander. And when that monarch addressed him with greetings, and asked if he wanted anything, "Yes," said Diogenes, "stand a little out of my sun."It is said that Alexander was so struck by this, and admired so much the haughtiness and grandeur of the man who had nothing but scorn for him, that he said to his followers, who were laughing and jesting about the philosopher as they went away, "But truly, if I were not Alexander, I wish I were Diogenes." and Diogenes replied "If I wasn't Diogenes, I would be wishing to be Diogenes too"' (Alexander 14). In Greek, Diogenes' words: "ἀπὸ τοῦ ἡλίου μετάστηθι"

Cicero says Diogenes said: "Now move at least a little out of the sun" (Tusculanae disputationes 5. 92) In Latin: "Nunc quidem paululum a sole."

Valerius Maximus: "To this later, for now I just want you not to stand in the sun" (Facta et dicta memorabilia 4.3. ext. 4). In Latin: "Mox ... de ceteris, interim velim a sole mihi non obstes."

OK, so it seems like the accounts are broadly consistent, although there's a lot of wiggle room in what exact words were said. (We could find some other inconsistencies among the stories -- such as the location of the alleged meeting, but the possibility that they met at Corinth in 336 BC is the likeliest.)

If the meeting did take place, the record of it was probably left to posterity by means of Onesicritus, who was a Cynic philosopher and a follower of Diogenes who later joined Alexander. Presumably it was due to him that we have the story: if the story is real, then Onesicritus might have been there or heard about it from his fellow-minded Cynics who were there, and then Onesicritus made sure Diogenes was immortalized.

I've already reported Fox's view that it happened, and we could find other historians, even as far back as the mid-19th century (e.g., G.E. Lynch), who think it didn't happen. I think that it isn't very decisive that there are so many, similar reports: such agreement might usually be striking, but it is surely possible that the stories are all derived from a common report. Many of the biographers (such as Diogenes Laertius) just repeat literally every story out there and leave it to the modern-day reader to sort out what's true. (This is a general style of ancient history exemplified probably chiefly by Herodotus; examples on the other side include Thucydides.) A similar fact is that Diogenes and Alexander became linked in the minds of ancient biographers and historians. I already said above that legend had it that they died on the same day, for instance. Was this connection there because the two met and got along famously, or is the meeting just a legendary story, as apocryphal as their having died on the same day? The reason why the meeting is such a powerful image isn't because it is allegedly historically accurate but because Diogenes and Alexander represent such different things, and Diogenes' eccentric lifestyle made him a popular subject for biographers and story-tellers. The fact that his follower Onesicritus joins Alexander makes it easier to join the two in people's minds (i.e., perhaps Onesicritus was spreading a totally made-up story).

It also bears clarifying that Diogenes was known for having many... colorful interactions. The one with Plato stands out: he is said to have picked up on Plato's definition of a human being in the Statesman and sought to give an embarrassing counter-example by showing that the definition is also satisfied by a plucked chicken. (This presumably happened while Diogenes was in Athens, where Plato was based.)

Further, I think a crucial feature of the story is, as I said above, that Alexander wasn't particularly special in 336 BC. This really tells against Diogenes Laertius' account. I quoted his account from VI.38 earlier, but at VI.60, he provides another telling that has Alexander introducing himself with "Alexander the great king," and Diogenes quips back "I am Diogenes the dog." This sort of exchange is meaningful to the reader only because we project backwards our knowledge of who Alexander became.

Just consider, for instance, Plutarch's story: the story works only because Alexander was looking for congratulations from everyone. He doesn't hear from Diogenes, so he seeks him out. The point of this is the reversal: the "great king" seeks out Diogenes. But it just isn't as powerful when we remember that, in fact, Alexander just wasn't "the Great" yet.

In conclusion, then, the meeting between the two is definitely possible, and there are a lot of very similar accounts of the meeting, but we don't really have any decisive evidence.