Even if you subscribe to the idea that Alexander can be seen as a "Persian king," he certainly wasn't the last one. There were several Persian royal dynasties throughout history, all of which - apart from the Achaemenids - ruled at various points in time after the conquests of Alexander the Great, so the idea that he was the "last Persian king" doesn't really work out. And that's if you see him as a Persian king. I don't, because he wasn't Persian. Yes, he kept many of the basic institutions of the Persian Empire intact - but he was decidedly non-Persian so I'm not too sure what use there is in claiming him to be Persian. The Parthians aren't really considered Persian despite ruling much of what was classical Persia, and the Parthians were far closer to the Persians, culturally speaking, than Alexander ever was.

As for being the "first Hellenistic king," I suppose one can see him as that if by "Hellenistic" they are referring to what is commonly known as the Hellenistic period, a period of Greek domination in the Eastern Mediterranean and Near East. He certainly wasn't the first Macedonian or Greek king, but thanks to Alexander, Greek culture went from being mostly isolated in Greece itself and the various Greek outposts of the Mediterranean and Asia Minor to being the ruling culture of much of the known world. Now, some people date the Hellenistic era as starting with Alexander's death, but if you move the date back a little bit to cover Alexander when he was still alive, one can make an argument for him being the "first Hellenistic king."

Frankly, I haven't heard either terms be used to describe Alexander the Great.

If you remove the actual individuals involved and just look at the political situation, you had the unified Achaemenid empire followed by the multiple Hellenistic kingdoms of the Diadochi. The political shift took place after Alexander’s death, so from a political (rather than cultural) perspective it might make more sense to group Alexander with the Achaemenids.

From the point of view of (say) a typical Egyptian farmer who never met Alexander or Darius and didn’t particularly care what nationality they were or what dynasty they belonged to, the relevant change came when Egypt stopped being a satrapy of a much larger empire and became an independent state with local rulers. Such a commoner would be more likely to remember Alexander’s rule as the end of the Achaemenids rather than the beginning of the Ptolemids.

There are convincing arguments for both, in my opinion, but I think calling him a the first Hellenistic king is more accurate.

Obviously he conquered the territory that had previously been controlled by the Persian Empire, but then much of that territory was subsequently (largely) inherited by the Seleucids. Alexander did try to blend Greek/Macedonian ideals with Persian ones, elevating Persians to high office (which dismayed many of his Macedonian officers), adopting many of the royal trappings of a Persian king, and certainly styled himself as a Persian ruler. I wouldn't disagree with someone calling him a Persian King - but if you think he was, then he wasn't the last.

As for the first Hellenistic king, I think he probably was. He had all the hallmarks of those that followed him (the dynasties created by his warring generals following his death) - he held Greek culture in very high regard but understood the importance of seeming in-tune with local culture, he placed a very high worth on military prowess, saw himself initially as the son of a God (son of Zeus or Amun) and later demanded to be worshipped as a God himself (instructions he sent to Greece), and styled himself as the saviour or protector of Greece. All of these things would come to define the Hellenistic age - plus, he conquered the territory that the Hellenistic kings would rule. So yes, first Hellenistic king, definitely.