Short answers, pulled from the story.
The Roman Kingdom began around 753 BC with the founding of Rome on the Palatine Hill and ended around 509 BC when the last king, Lucius Tarquinius Superbus, was expelled. The period lasted roughly 243 years according to the traditional chronology codified by Varro.
Seven kings ruled the Roman Kingdom in succession: Romulus, Numa Pompilius, Tullus Hostilius, Ancus Marcius, Lucius Tarquinius Priscus, Servius Tullius, and Lucius Tarquinius Superbus.
When a king died, the Senate appointed an interrex to nominate a candidate within five days. The nominee was approved by the Senate, then voted on by the Curiate Assembly. The king-elect still had to receive divine confirmation through augury and formally propose a law granting himself imperium before taking power.
The kingdom ended in 509 BC after Sextus Tarquinius, son of King Tarquinius Superbus, raped Lucretia, who then took her own life. Lucius Junius Brutus led a revolution that expelled Tarquinius and his family. Brutus and Lucius Tarquinius Collatinus became the first consuls of the Roman Republic.
The Roman king held imperium for life, giving him supreme military, executive, and judicial authority. He was commander-in-chief of the legions, chief judge, chief priest, and the only person who could appoint senators or perform augury on behalf of Rome. The sole act requiring Senate and assembly approval was declaring war.
The Gauls sacked Rome after the Battle of the Allia in 390 BC according to Varro, destroying most historical records from the regal period. What survived was eventually lost to time or theft. All surviving accounts were written centuries later by authors such as Livy, Plutarch, and Dionysius of Halicarnassus, drawing largely on oral tradition.