The story of Perses deposing Aeëtes and getting killed by Medea can be found in Apollodorus 1.9.28:
Medea came to Athens, and being there married to Aegeus bore him a son Medus. Afterwards, however, plotting against Theseus, she was driven a fugitive from Athens with her son. But he conquered many barbarians and called the whole country under him Media, and marching against the Indians he met his death. And Medea came unknown to Colchis, and finding that Aeetes had been deposed by his brother Perses, she killed Perses and restored the kingdom to her father.
Apollodorus. Apollodorus, The Library, with an English Translation by Sir James George Frazer, F.B.A., F.R.S. in 2 Volumes. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1921. Includes Frazer's notes.
Hyginus gives a slightly different account of his death in Fabulae 27. Here it is Medea's son Medus that kills Perses:
An oracle told Perses, son of Sol, Aeetes' brother, that he should beware of death from Aeetes' descendants. Medus, following his mother, was brought to him by a storm, and guards seized him and brought him to King Perses. When Medus, son of Aegeus and Medea, saw that he had come into the power of his enemy, he falsely asserted he was Hippotes, son of Creon. The king carefully investigated, and ordered him cast into prison. There sterility and scarcity of crops are said to have occurred. When Medea had come there in her chariot with the yoked dragons, she falsely claimed before the king to be a priestess of Diana. She said she could make atonement for the sterility, and when she heard from the king that Hippotes, son of Creon, was held in custody, thinking he had come to avenge the injury to his father ... there, unknowingly, she betrayed her son. For she persuaded the king that he was not Hippotes, but Medus, son of Aegeus, sent by his father to dispatch the king, and begged that he be handed over to her to kill, convinced that he was Hippotes. And so when Medus was brought out to pay for his deceit by death, when she saw that things were otherwise than she had thought, she said she wished to talk with him, and gave him a sword, and bade him avenge the wrongs of his grandfather. Medus, at this news, killed Perses, and gained his grandfather's kingdom; from his name he called the country Media.
In Library Diodorus Siculus informs us that Perses was king of the Tauric Chersonese and that he fathered Hecate, conflating him with the Titan Perses, son of Crius. Diodorus also has Medus killing Perses instead of Medea:
4.45.1. Since it is the task of history to inquire into the reasons for this slaying of strangers, we must discuss these reasons briefly, especially since the digression on this subject will be appropriate in connection with the deeds of the Argonauts. We are told, that is, that Helius had two sons, Aeëtes and Perses, Aeëtes being king of Colchis and the other king of the Tauric Chersonese, and that both of them were exceedingly cruel.
4.45.2. And Perses had a daughter Hecatê, who surpassed her father in boldness and lawlessness; she was also fond of hunting, and with she had no luck she would turn her arrows upon human beings instead of the beasts
4.56.1. Speaking generally, it is because of the desire of the tragic poets for the marvelous that so varied and inconsistent an account of Medea has been given out; and some indeed, in their desire to win favour with the Athenians, say that she took that Medus who she bore to Aegeus and got off safe to Colchis; and at that time Aeëtes, who had been forcibly driven from the throne by his brother Perses, and regained his kingdom, Medus, Medea’s son, having slain Perses; and that afterwards Medus, securing the command of an army, advanced over a large part of Asia which lies above the Pontus and secured possession of Media, which has been named after this Medus.
Diodorus Siculus. Diodorus of Sicily in Twelve Volumes with an English Translation by C. H. Oldfather. Vol. 4-8. Cambridge, Mass.: Harvard University Press; London: William Heinemann, Ltd. 1989.
I couldn't find a source directly connecting Perses to Persia. I assume Madeline Miller is conflating Perses of Colchis with Perses, son of Perseus. The latter was identified as the ancestor of the Persians by several writers (e.g Herodotus and Plato), having fathered the Perseides with the Oceanid Perse. Which of course - if we assume two different Perses - would make Perses of Colchis a half-brother to first generation Persians.
Hat tip to b a for noticing that German Wikipedia identified its sources.