The speedy campaign to conquer the Persian Empire poetically saw the equally speedy collapse of Alexander's Empire. Unlike, say, the fall of the Roman Empire, we can more easily diagnose the issues of why it had quickly fallen into the Wars of the Diadochi.
- The lack of a viable heir. The Argead dynasty had ruled for generations, albeit with many ups and downs as the violent Macedonian court politics tended to periodically cull the number of male heirs in the family. When Alexander died, he wasn't left with a clear line of succession: Philip Arrhideus, Alexander's half-brother, was considered mentally handicapped (either intellectually disabled or severe epilepsy); Alexander's son by his Sogdian wife Roxana was not yet born, so they did not even know if it would be a male heir - never mind that he wasn't of full Macedonian stock; Heracles, an alleged bastard fathered by Alexander and a Persian noblewoman named Barsine, is not even fully agreed upon by historians to be a legitimate claim. No male heir = no one to take control of the army = opportunistic courtiers and generals attempting to manipulate the scenes
- The size of the empire, and Alexander's failure to consolidate/administrate. Alexander justifiably deserves his reputation as an amazing general and military thinker. But in terms of administration of the empire, he rarely innovated unless he absolutely had to. With some credibility, Alexander has been called the last Persian ruler, because he attempted to insert himself into the machinery of Persian governance and administration. Unfortunately, Persian rule was based largely on personal relationships with the political elites of each area of the Empire, and Alexander had heavily disrupted this network. The empire's sheer size was also a major problem, as by the time he got back to Babylon his easternmost satrapies in India had already revolted or broke down. Many of these lands had yet to be fully explored, and to try and actively maintain such wide geographic area in the time of the horse was a logistical nightmare - a problem recognized and felt by the Seleucid dynasty, who took over much of the Near East and Central Asia. If given time, I am sure Alexander could have reworked or better integrated his administrative system, as did both Ptolemy in Egypt and Seleucus in Mesopotamia/Syria. But as far as our sources suggest, Alexander had no plans of settling down, and instead was planning new expeditions even from his deathbed.
- The nature of Macedonian kingship. Following #1, Macedonian kingship had largely been restrained to the Argead dyansty. But the unique circumstances of Alexander's death and lack of an heir had created an environment where kingship was less determined by relationship to the royal house, and more about two important rules: 1) You were able to prove your right to kingship and kingdom via conquest and victory in battle (often phrased as "spear-won land"), 2) Being able to pay for armies with vast amount of money and treasure, something that was in plenty supply after plundering the treasuries of the world's then-richest empire.
- The Main Players of the Wars of the Diadochi: We need to take in consideration of the main actors of the Wars of the Successors. These were men who were battle hardened commanders, extremely talented/experienced and equally as ambitious. Some of them were much more antagonistic towards the remnants of the Argead house, such as Cassander or Antigonus the One-Eyed, both of whom contributed to the extinction of the house by killing of its remaining members (especially Cassander, who murdered Alexander's mother, wife, and son). Many of these men grew up with Alexander, or were in his personal circle of advisers and friends. But even when Alexander was alive, we see glimpses in our sources of the political infighting and rivalry of these men who saw each other as competitors in the great game for power and wealth.
These are the biggest and most obvious contributors as to why the empire of Alexander collapsed, at least in my opinion. I highly recommend you check out either Robin Waterfield's "Dividing the Spoils" or James Romm's "Ghost on the Throne" if you want to get a more in-depth look at the Wars of the Diadochi