This is a long and very convoluted story with seemingly no end. The new world order of the Hellenistic successor kingdoms did not completely fall into place until the mid 270s, more than forty years after Alexander died. When the conquering hero died in 323 he had left no feasible plans for how the empire would be governed and there was a complete lack of central authority. Even when Alexander was alive the fact that he was always off somewhere fighting someone meant that the territories behind him operated more or less autonomously. Alexander's empire was held together through sheer force of will by Alexander himself so when he died his exceedingly ambitious officers immediately began to fight over it.

So how did it all happen? As I said it's a long story but I'll try and give a readable summary. First of all Alexander didn't leave an heir when he died, perhaps because the dynastic gene pool had run pretty dry by 323. Through a great deal of bickering an agreement was reached whereby Alexander's mentally handicapped half-brother Philip III became king jointly with Alexander's infant son Alexander IV (who was unborn when Alexander died but became king when he entered the world). Not everyone had wanted members of Alexander's family (the Argeads) to take the throne but many were loyal to the dynasty (or at least said they were) and wanted an Argead on the throne. A general named Perdiccas served as regent for the two kings-in-name-only and the empire was divvied up. Babylon was to be the capital

Things went off the rails immediately. Perdiccas wasn't happy with just the regency and perhaps aspired to the crown himself. He conspired to marry Alexander's sister Cleopatra in an attempt to manufacture a royal claim and started a war. Antipater, the viceroy of Macedon, marched into Anatolia with a general named Craterus to face the would-be tyrant. Elsewhere, Ptolemy (who was already acting like an independent ruler) had orchestrated the redirection of Alexander's funeral procession to Egypt in his own bid to gain prestige. Perdiccas decided to invade Egypt and left a contingent to deal with Antipater and Craterus.

Perdiccas failed in his Egyptian campaign and was killed by his own men in 321, bringing an abrupt end to hostilities. The empire was redivided and a general named Antigonus was given the job of leading the cleanup operation against Perdiccas' old supporters. They were still numerous and were holed up across Anatolia. One of them, Eumenes had even been able to defeat Craterus when Antipater had invaded. Antipater was made the new regent and the kings went to Macedon with him.

Antigonus was able to deal with the remaining Perdiccans, scoring decisive victories over the generals Eumenes, Alcetas, Attalus, Docimus, and Polemeon. In the late summer of 319 Antipater died but left the regency to an old friend of his named Polyperchon. Antipater's son, Cassander, was angry at this and overthrew Polyperchon with Antigonus' help. I'm skimming over a lot here but by 316 Cassander had managed to reduce Polyperchon to a handful of outposts in the Peloponnese and ruled Macedon for the boy king Alexander IV (Philip III had been assassinated in the convoluted web of intrigues that I'm glossing over).

While all this had been happening in Europe Antigonus had been busy in Asia. Having helped install Cassander he then had to deal with Eumenes. This Perdiccan had reneged on an amnesty Antigonus had offered him following his defeat and had gone on the run, raising an army along the way. Antigonus gave chase and the two spent the next three years running around the Middle East fighting each other. Eventually, in 316, Eumenes lost the Battle of Gabiene by treachery and was executed by Antigonus. In the course of the war the latter had gained a great deal of wealth and influence in Asia. He made sure that all the governors of the eastern provinces were loyal to him and did away with the ones that perhaps weren't. One of these governors was Seleucus in Babylon, who made himself scarce and fled to Ptolemy. When Antigonus came back to the Mediterranean he faced a hostile coalition of Ptolemy, Cassander, and Lysimachus (governor of Thrace). Antigonus was too powerful for their liking and they resolved to stop him.

From 315-312 the two sides fought an indecisive war in Greece and Anatolia. No breakthrough occurred on either side until Seleucus persuaded Ptolemy to invade Palestine and hopefully help him get to Babylon and reclaim his province. The invaders scored a decisive victory over Antigonus' volatile son Demetrius at the Battle of Gaza in 312 but couldn't turn it into a knock-out blow. Demetrius organized a strong counter attack while his father came with reinforcements. Ptolemy didn't want to risk a showdown and withdrew, but not before sending an insistent Seleucus on his way to Babylon with 1000 men into the desert with winter setting in. Ptolemy must have thought it was a suicide mission.

In 311 Antigonus and the coalition made peace that left things more or less status quo anti bellum. But the peace didn't last as Ptolemy quickly regrouped and set about harrying southern Anatolia. He even attempted to marry Alexander's sister and pull off what Perdiccas had attempted but to no avail. Cassander quietly did away with Alexander IV and tried to induce Antigonus' allies in Greece to defect to him. Seleucus meanwhile had overcome the odds and had not only marched into Babylon in mid 311, he had routed the numerically superior force that had come to stop him. Demetrius was sent to deal with Seleucus while Antigonus neutralized Ptolemy's pursuit of Cleopatra by having the woman murdered. But Cleopatra had been the least of his problems. By 309 Seleucus and his supporters had managed to repel Demetrius so Antigonus went to do the job himself. He failed and the two agreed a border somewhere near the Euphrates. Antigonus had lost his huge Asian holdings but was now in possession of a much more governable domain comprising Anatolia and Mesopotamia. Seleucus went on a grand expedition to claim the rest of Alexander's old eastern conquests but we have almost no record of it.

War continued in the west. In 306 Demetrius smashed Ptolemy's navy off Cyprus and Egypt was left vulnerable. Following the victory Antigonus did away with the pious fiction that he was ruling in the name of the Argeads (that had had been the official line since the 311 peace and before) and proclaimed himself king. The other dynasts quickly followed suit. He then attempted to invade Egypt late in 306 but was unable to break through, thus deciding to leave in order to avoid Perdiccas' fate. He pulled back but Demetrius went about unsuccessfully besieging Rhodes in 305/304 before going to Greece to build more support there and annoy Cassander.

In 302 Lysimachus decided to go all in. He planned an invasion of Anatolia from Thrace and persuaded Ptolemy, Seleucus, and Cassander to help him out. The result was the Battle of Ipsus in 301. There the 81 year old Antigonus led his men against his old enemies in a decisive engagement. The old man lost the battle, his kingdom, and his life. Demetrius escaped, having been drawn off the field in a bout of overenthusiasm and quickly barred from returning by Seleucus' elephants, freshly acquired from India.

The Antigonid territories were divvied up with Lysimachus coming out the best, getting his hands on Anatolia. Ptolemy had not made it to the battle, but nonetheless got control of Palestine (he'd invaded the territory in 302 but hadn't gone to Ipsus). This annoyed Seleucus but he let it slide. Demetrius was left as a king with no kingdom but still had a considerable navy and so began life as a sea king.

Demetrius' time as a roving warlord came to an end in 294 when he seized Macedon from the feuding sons of Cassander (who had died a few years earlier), greatly alarming the other kings. He reigned for a wild six years until he was forced out of Macedon by Lysimachus and Pyrrhus (he of the Pyrrhic victory against Rome). Deprived of much of Macedon and Greece, he left his son Gonatas in the handful of cities he still had and hurled himself into Asia for one last adventure. It didn't go well. Lysimachus completely outdid him and he wondered into Seleucus' territory where he was captured and placed under house arrest. He drank himself to death in 285.

In 281 Seleucus and Lysimachus (who were both at least 70 by this point) went at in one last successor war. Why they fought is a tangled web of intrigue that I don't want to get into but at the end of it all Lysimachus lay dead on the plains of Corupedium. Seleucus was now the last successor standing. Ptolemy had died in 283 but Seleucus was still going strong. He was on the verge of uniting Alexander's European and Asian conquests, only Egypt remained out of his grasp. He marched into Thrace, celebrated a bit (I imagine) and was promptly murdered by one of his officers. The world plunged into chaos once more.

Out of this chaos emerged the last pieces of the new world. Gonatas eventually took Macedon while Asia Minor was split between Seleucus' son Antiochus and various independent native fiefdoms. So by 275 or so the new order was in place. The Seleucids held most of Asia, the Ptolemies held Egypt and Palestine, a patchwork of regional strongmen ruled much of Anatolia, while the Antigonids held Macedon. These kingdoms would endure and presided over a flourishing of Greek culture across the east.

I hope all that was somewhat informative and I apologize in advance for glossing over so much. It's just very difficult to fit everything into a digestible reddit post so I hope you got the gist.