They decided by fighting. A lot. They fought for over forty years (approximately 323-280 B.C.) before any sort of stability arrived. These wars are called the wars of the diadochi (successors).

It's a long and complicated story, but to cut it short: Alexander left behind two dynastic successors: A half brother named Philip Arrhidaeus who was left behind in Macedon and who seems to have been handicapped in some way, and a son named Alexander (IV) who was born after Alexander the Great's death. Initially, then, none of Alexander's generals declared themselves kings, but fought over the regency and individual territories; the regents who came to power assigned territories to some, and others just took them. One example of the latter is Ptolemy, who stole Alexander's corpse and established his power in Egypt. In any case, alliances and regencies shifted so often that no single general ever attained any real legitimacy or was able to enforce a long lasting peace.

By 310 Alexander's half-brother (Philip Arhidaeus, killed by Olympias), mother (Olympias, killed by Cassander), son (Alexander IV, killed by Cassander) and wife (Roxane, killed by Cassander) had all been killed off, ending his dynasty. In 306 Antigonus the One Eyed and his son Demetrius were the first to take the title of king, and the other successors soon followed and began establishing kingdoms. Over the years Ptolemy retained control in Egypt, Seleucus eventually took control of most of Alexander's Asian empire, and Lysimachus attained more and more success in mainland Greece, Macedonia and the Balkans region. At the battle of Corupedium in 281, Lysimachus and Seleucus fought and Seleucus won. At this point, Seleucus essentially controlled all of Alexander's empire except for Egypt. Unfortunately for him, soon after the battle the disowned wild card son of Ptolemy, Ptolemy Keuranos (The Thunderbolt) assassinated Seleucus. Power over Macedon and Greece fell to Demetrius and his son Antigonus Gonatas, while Seleucid's family continued to rule in Asia and the Ptolemies enjoyed their Egyptian kingdom. These three dynasties/monarchies would remain relatively stable for the next 100 years or so, until Rome became more heavily involved in the Greek east. Of course, other more minor kingdoms also arose in the Greek world, many Greek city states remained autonomous or formed federal leagues, and the kingdoms fought incessantly and gained and lost territory in the two centuries after 280.

It's a fascinating period and I can't do it justice on reddit. If you're interested in learning more about these wars, there have been two recent books published that are written more for the layman reader: Robin Waterfield's Dividing the Spoils, which covers the whole period, James Romm's Ghost on the Throne, which covers the first fifteen years or so. Primary sources to check out include Diodorus Siculus' Library of History, books 17 onward or so, though there's not really an affordable translation out there for this. A better bet for you might be the penguin edition of Plutarch's biographies "The Age of Alexander", which contains lives of prominent personalities of this time, including a few of the successors.