I ant speak for Genghis Khan, but I'll address the Alexander question in two folds:

Why did Alexander push so far?

Quite literally he wanted to conquer the world. There are some debates to whether Alexander truly viewed him self divine on Earth, but he certainly believed himself the son of Zeus. You would have to remember who Alexander's role models were, the Hercules and Achilles' of the world. Note both are fictional character with larger than life ambitions. This rubbed off on Alexander; he not only wanted to live up to his divine destiny but to also do something so glorious he would be remembered as Achilles and the likes were.

The second question, how did he justify this to his troops?

Well in several ways. The universal theme through his conquest was glory and riches (typically a universal motivator in most of warfare.) After the Macedonian invasion of Persia, Alexander became by far the richest man on Earth and his Macedonian soldiers raked in big time too. Also it is important to remember that for long stretches of time Alexander used Mercenaries who often will fight anyone for money.

Outside of those two factors, he used motivators that applied to circumstance;

-He invaded Persia by declaring revenge on past Persian atrocities to the Greeks. Naturally the Greeks got behind this.

-After Darius is murdered by his own men, he launches a new invasion eastward to "capture those who killed Darius." You can debate if he actually cared, but in any event many Persians got behind another revenge war.

-After mainland Persia, much of Alexander's Persian campaigns was to consolidate his power/hold. What good was all that conquering and wealth if you couldn't hold on to it. It is at this point that Alexander begins to adopt more Persian customs, to the delight of Persians but the dismay of Macedonians/Greeks.

-Alexander turns further East towards India. At this point, intoxicated on immense success and eternal glory, Alexander wants to lead his men on the creation of a one world kingdom under his leadership (keep in mind India was probably the furthered the the Greeks got in the East.)

It is at this point that his men rebel, especially his Macedonians. They had fought bravely and loyally to him for easily over a decade. Furthermore he had along the way promised them they would eventually return home. These men viewed the Non-Greeks as barbarians and saw them getting more attention from Alexander. By this point there was nothing Alexander could say or do. They wanted to go back to their families, bask in their glories at home and enjoy their wealth

Please feel free to ask follow up questions.

Source: Alexander by Robin Lane Fox