Genghis Khan's empire did not collapse upon his death, but only expanded more, it was not until the death of Möngke Khan, more than 30 years and a couple of Great Khans later, that the empire broke apart.

Empires require a line of succession to keep control within the hands of one family, or line of rulers.

Alexander's empire was split up between his generals, decentralizing the political power. In addition, the military was split, leaving the empire open to rebellion and revolution throughout the territories it controlled.

The Mongolian empire was different. As /u/DeSoulis pointed out, it lasted more than three decades after Genghis' death. This was because Genghis foresaw the issue and named Ogedai as his successor before he died. He also had his generals swear to honour his wishes, and so the empire passed without trouble to Ogedai. The other factor that made this transition easy was that Genghis had the loyalty of his brothers, and of his sons. By the time of Ogedai's death, there were multiple sons and nephews who could claim direct lineage, and so the succession was more contentious. This was the cause of the horde's return from Europe, where they had just destroyed the Hungarian armies.

The key to the survival of these empires is maintaining a clear line of succession. There was no rule among the Mongols for who succeeded the old Khan - it was done by nomination or by garnering support. Alexander died relatively young, and therefore was possibly unprepared for planning the handover of an empire.

On the other hand, the Tokugawa dynasty in Japan(not an Empire, but still displays the importance of succession), was planned out by the first shogun, Ieyasu, so that his heir was nominated during his lifetime, succeeded and began ruling during his lifetime, AND was approved by the emperor - in effect triple sealing his right to rule Japan.

You're phrasing it in very broad terms, but you're not really wrong.

New polities are inherently fragile. "Great empires forged by single men" by definition unite many people in the same state through massive conquests, and the places united are often unhappy with the people who conquered them, who tend to set themselves up as a new ruling class and defeated them in war. And war itself often brings about its own reasons for anger - the Mongols would massacre cities which refused surrender as a terror tactic, while Alexander famously burnt Persepolis, the historic Persian capital.

Both Alexander and Temujin (often known by his title of Genghis Khan) made efforts to integrate the conquered into their new states - Alexander instituted proskynesis, adapted many Persian customs, and insisted his officers marry Persian women, while Temujin tolerated and patronized all religions within his realm and allowed some skilled individuals from peoples he had conquered to reach significant positions of power. These efforts worked, but only to a point.

The new ruling class itself is often unstable - they're ruling over a lot of land and people they never governed before, leading to social changes of their own, and efforts to win the loyalty of newly conquered peoples often alienate traditionalists in their own armed forces/ruling class/ethnic group. (i.e. Kublai Khan's efforts towards Sinicization alienating traditionalist Mongolians who supported Ariq Boke in that civil war.) Conquest at minimum means new frontiers and opportunities, but this can itself break down prior social structures, and in the case of Alexander and Temujin conquest was preceded by unifying the people who would be doing the conquering. And people react to those social changes in many different ways even within a ruling class - consider Hulegu Khan declaring war on the Abbasid Caliph in another campaign of conquest, while his cousin Berke, also a Genghisid descendant and ruling a different part of the empire, had adopted Islam and declared war on him in response!

These factors exist during periods of expansion, but the death of a great conqueror removes the person who held the empires together and had so long managed to paper over these differences, whether through fear, reputation, a string of victories (which surely pleased those looking for plunder and land) or personal charisma. In doing so it often provides an opportunity for both fighting over who gets to replace them and opportunities to create breakaway states, processes which often feed into one another - if the army's fighting over who runs the country they're not suppressing an independence rebellion in its distant corners.

Note that this collapse isn't always immediate, some expansionist states avoid it outright, and even in the case of Alexander and Temujin (and the Mongol Empire didn't really fragment until a generation and four khans later, although you can certainly find echoes beforehand) some successor states survived centuries, albeit not in the empire's full borders - the Crimean Khanate was ruled by Genghisids until 1783, and Egypt remained under the rule of a family descended from one of Alexander's generals for nearly three centuries. But newly formed empires often have significant internal divisions, and the process required to forge a loyalty to the empire in itself takes time and often a lot of bloodshed.