The kind of succession crisis that Alexander III of Macedon faced when he ascended the throne was not common in Ancient Greece, because hereditary monarchy was an extremely rare form of government among the Greek states. Only a handful of them had "royal" families at any point, and most of these were hereditary tyrannies that would only last for a few generations at most. In the wider Greek world there were a few stable monarchies (for example those of the Crimean Bosporos and Cyrene in Libya). But on the Greek mainland only Sparta persisted with its peculiar dual monarchy (i.e. they had two kings).

The Spartan succession was often messy, with our sources reporting numerous regencies, manipulations of the line, and contested cases. Often there were several potential claimants among the current kings' extended families. A number of prominent Spartan kings only came to power because the more likely candidates had been exiled or killed. For instance, Leonidas was only the 3rd eldest son of Anaxandridas, and ascended the throne when he was already 50. When he got himself killed at Thermopylai, his son Pleistarchos, still a boy, became king; Pausanias ruled as regent in his name.

Even so, the Spartan succession rarely involved open violence against rival claimants. The reason Leonidas became king upon the death of his half-brother Kleomenes was that his other half-brother Dorieus had simply opted to leave Sparta when his brother became king. Leonidas himself served quietly as a regular Spartiate under king Kleomenes for most of his adult life, as his own twin brother Kleombrotos would do for him. The royal families would no doubt have lost much of their legitimacy if they made a habit of murdering their potential rivals for the throne.

In short, Greek monarchy (where it existed) did not normally feature a royal killing spree whenever a new king came to power. What made Macedonian monarchy so different?

One reason must be the simple fact that, compared to Greek monarchies of the time, the fourth-century BC Macedonian kingdom was very unstable. At the time of Philip II's accession in 360 BC the state was on the brink of collapse. Kings had succeeded one another rapidly for decades; there were 12 kings in the 40 years before Philip, often with outside agents like Athens, Sparta and Epiros pulling their strings. These kings were as concerned with quelling internal rebellions as they were with holding invading enemies at bay. Philip became king of a country in ruins, threatened from all sides and from within. We are not very well informed about the power struggles of this tumultuous period, but it's fair to assume that the remnants of the factions of some of Philip's rivals would still have been around in 336/5 BC, waiting for their chance to renew their bid for the throne. They may even have been behind his assassination (though there is no evidence for this). Any lingering allegiances to old rivals of Philip would have been a great excuse for Alexander to get rid of some potential troublemakers.

The other reason is that Philip pursued a very successful policy of arranging diplomatic marriages to tie his elite supporters and neighbouring rulers more closely to himself. This was effectively impossible in the Greek world, where monogamous marriage was the norm. Tyrants like Dionysios of Syracuse, who kept more than one wife, did much to cement the stereotype that tyrants were by nature perverted, sexually insatiable degenerates. But in Macedon royal polygamy was apparently not considered a problem. By the end of his reign, Philip II had at least 7 wives (there may have been more). Alexander was his first son by his fourth wife, Olympias.

Effective as these marriage ties may have been as a way to seal alliances and keep powerful families mollified, you can imagine the crises they created when a ruling king died. Much of the ongoing chaos of early 4th-century Macedon was caused by the existence of several branches of the royal family all vying for power. Conflict erupted again when Philip II was murdered, with several of his children claiming the right to rule. Alexander no doubt felt compelled to assert his authority both through victories in battle (against the Illyrians, Thracians and Thebans, all within the first year of his reign) and through a forceful statement toward anyone with thoughts of rebellion. This was the reason behind some of the more outrageous acts he committed against Macedonian elite families early on.

Even so, he was unable to put all rivalries to rest; upon his death, succession conflicts broke out again even in Macedon (to say nothing of his empire as a whole). While much of the struggle revolved around seizing guardianship of Alexander's wife Roxane and their infant child Alexander IV, his half-brother Philip Arrhidaeus was equally important - and then there were the various clans of other people who had married into the Argead house. With such a family tree, succession crises were almost inevitable, and the kings of Macedon would often resort to violent measures to assert their claims.