Royal cult developped in dynasties of generals of Alexander the great, but there are only pieces of information about their creation. The main elements are that the first sacrifice to a living man in a greek city comes from the early 4th century BC and so took place a few decades before Alexander's conquests. The second element is the importance of the body of Alexander. It was not necessarily seen as divine but was an important element of legitimacy in the fight between the generals. That's why Ptolemy highjacked the corpse to bring him to Alexandria next to Egypt (The Alexandria) to show that he is the rightful heir of the empire. He put it into a golden sarcophagus in some kind of mausoleum in the palatial district of the city.

But Egypt didn't had a monopoly on the practice. In the rival seleucid kingdom (from Seleucos, another bodyguard of Alexander) there was also religious acts in his honor, but it seems less organized. That may only be a bias of our sources, much more abundant in Egypt. We know from the inscription OGIS 222 (Ogis stands for Orientis graeci inscriptiones selectae) that while taking hommage to the king Antiochus I (the son of Seleucus) the inscription mention a ritual to honor Alexander. But didn"t remember to read anything about a priest of Alexander in the seleucid kingdom. The best documentation on the seleucid royal cult came from the organization of the cult of Antiochus III and his ancestors and Alexander isn't in it. It came partly because it could outshine the "heroic" deeds of Seleucus, the builder of the Syrian Tetrapolis, tamer of (tribes of) the Taurus (mountains). To achieve that goal the Seleucids renamed places (while refounding it like the Alexandrias of the East attacked by nomads and renamed Antioch + [insert river, mountain or region name, ideally river] and rewrote history. One good exemple of the later is that the restitution of the Tyrannochtons (tyrant killers) a group depicting the killers of Pisistratus the last tyrant of Athens and thus symbol of the Athenian democracy. The statue is told to have been taken as a trophy by the persians in 480BC during the persian wars. Then there are various accounts saying that a king bring it back. Some say Alexander, the other Seleucus and Antiochus and the later to an Antiochus. The difference is huge as it put the accent on the good deeds of the seleucid dynasty. In my opinion it's a manipulation from Antiochus III in a charm offensive on the greek cities in order to find allies (or at least keep them neutral) while he was "liberating greece" between 192 and 188 BC, but we can't be certain.

The seleucid royal cult survived for a surprisingly long time as there is a Parthian document mentionning the priest of the seleucid dynasty in the second century AD, long after their end.

Now to return to Alexander several emperors visited the tomb and it is said that Augustus broke his nose while kissing him. To be honest I don't know if the cult continued in the same organization as under the Ptolemies and I don't have access to Jstor right now so there is a first page teasing me of the rest... Don't forget that other kings and romans tried to reuse Alexander for their own prestige, like Mithridates VI of Pontus, Pompey "the great" or even Caracalla who even recreated a phalanx to parade like the macedonian