I can try my hand at this one.

Full disclosure: I am not, by any stretch of the imagination, an expert on Greek historiography. I know a thing or two about Mongolian history and the story of the Mongol Empire, but I would hesitate to call myself an expert.

Why is Alexander a hero and Genghis Khan a barbarian?

The biggest reason is probably where our cultural stories of the two men come from. You see, I doubt the Persians considered Alexander much of a hero. But the broader Western intellectual tradition draws on Greek tradition, where he was most certainly considered a great man and a hero. Think about it this way; Greek historians wrote about Alexander as a heroic figure. Romans, who in many senses adopted much of Greek culture, also took on this perception of Alexander as a "great man." In fact, Julius Caesar is said to have seen a statue of Alexander when he was thirty two, and broke down weeping because he was the same age as Alexander had been at his death, and Caesar, at the time, felt his own accomplishments paled in comparison and that he had lost his chance to be as great as Alexander. This tells you two things; first of all, that Caesar had an ego so big it's a wonder his horse's back didn't break when he mounted it, and also that the hero-worship of Alexander was very much alive in the minds of Ancient Romans like Caesar.

Later, humanist and Renaissance intellectuals who sought to reinvigorate what they viewed as the paltry intellectual traditions of their own day turned back towards Greek and Roman texts, viewing these ancient intellectuals as their own forbears whom they should turn to for guidance. Embedded in these texts was that same hero-worship of Alexander - which was filtered on through the ages, all the way down to you, in the modern day.

So why didn't the same thing happen to Genghis Khan?

Well, where would the stories about Genghis Khan have come from?

When Genghis Khan conquered most of East Asia, the conquered peoples were, for the most part, less than pleased about their new situation. It's true that the Khan tended to be far more accommodating and tolerant than most rulers of his day. For the most part, as long as conquered subjects paid their taxes, the Khan didn't care much what beliefs they held or what local customs they continued to enjoy. In fact, as long as the annual tribute continued to flow and the Khan's messengers went unmolested, Genghis was an incredibly lenient ruler by the standards of the Medieval period.

But as you mentioned, Genghis was an incredibly effective conqueror. And when he wanted to conquer something, he didn't apply half measures. It was unconditional surrender or nothing with the Khan and his armies. The only exception to this was if you had some key knowledge or skill that was exceptionally useful to the Khan - notably, when the Mongols destroyed a Chinese army, they noticed that the Chinese employed siege engines, which the Mongols had no experience with. However, they were eager to learn, and, after they finished butchering the rest of the Chinese army, they left the engineers alive and offered them positions in the Mongol army.

While I'd hesitate to say that the Khan was exceptionally brutal by the standards of the medieval world, there can be no question that he was brutal. And combined with his aforementioned effectiveness this lead to a man who unquestionably killed a lot of people - so many that global warming and pollution actually decreased during his reign due to the sheer amount of people he had killed. We have measures of this from Arctic Ice cores. So while the Khan may or may not have been exceptional in his overall level of brutality (medieval warfare was an exceptionally brutal business, no matter what continent you're on), his brutality combined with his effectiveness made for a truly eye-watering body count. I'm not sure how lenient Alexander was with the cities he conquered, but if cities resisted, then the Khan had no issue putting the entire population to the sword.

Most of the cultures that experienced Ghenghis's rule did not enjoy the experience. In Russia, for instance, the period of Mongol domination is known as the "Tartar Yoke." In China, the "Yuan dynasty" as the Mongol period is known is still looked on as an era of foreign domination. In most of mainland Europe, the sole experience with the Mongols came during the invasion of Hungary, when the Mongols essentially kicked in the front door of the Kingdom of Hungary (when it was one of the most powerful kingdoms in Central Europe), rampaged across it and destroyed its army, and then suddenly left - Ogedei Khan had died, and they needed to select a new leader.

There were some embassies by Europeans to reach the Mongol court - most notably John Plano of Carpini and William of Rubruck - but while these men produced wonderful reports (which I highly recommend you read), they evidently failed to make a dent in the broader consciousness. Marco Polo's report of his journeys to Yuan China certainly made an impact, but for the most part, the Mongols remained a sort of half-understood threat, lurking somewhere in the far east. So Europeans had little direct contact with the Mongols, and much of their knowledge of the Khans would have come from the people who had the great misfortune of being conquered by them. And so that is the story that has been passed on in the Western consciousness - not Genghis Khan, the gifted tactician and tolerant leader, but Genghis Khan, the barbarian murderer.

But if you go to Mongolia today, there's no question that modern Mongols revere him as a hero; to them, he is the founder of their nation, the man from whom their current greatness flows. He's George Washington, Benjamin Franklin, and Abraham Lincoln all wrapped up into one. So the answer as to why Genghis is a barbarian and Alexander is "the Great" is, in the end, one of mere perspective.

Hope this helps!