We do have explicit evidence of the existence of kingship from the inscription on the Lapis Niger, which was discovered underneath the Roman Forum dating to the late 6th century BCE. On the stone was inscribed several lines, of which also contained the word "recei" which is the dative/ablative(?) case translating to "for the king", sorry my Latin is a bit rusty.

The 7 legendary kings of Rome from Romulus to the Etruscan dynasty that ended with Tarquinius Superbus would have had to have reigned on average ~35+ years each to span the gap between the traditional founding of Rome in 753 BCE to the establishment of the Republic in 509BCE. This is more of a reflection on the idea that the Romans did once exist under a regal period, something that is accepted by historians, but there were probably many more rulers which are historical who we'll never know about.

Early Roman kingship was almost certainly tied to a religious nature as can be seen from the preservation of titles such as 'rex sacrorum' (king of sacrifices) which survived into the Republic. In addition to this the chief priest of the Roman Republic was titled 'pontifex maximus' (literally 'chief bridge-builder), possibly alluding to a much older tradition of the religious authority's relationship with the god(s) of the Tiber River. One of the things these early kings would have had to take into consideration was the population of arms-bearing adult men, known as the comitia curiata, which formed the original popular assembly, and to which the king turned to for military service.

All in all to go back over your questions. The first concrete rulers can't be established for certain until the abolition of the monarchy and the institution of the republic, the former of which consensus says existed. Its historical workings were originally likely religious in nature, taking on the character of an absolute monarchy which eventually evolved into a more oligarchic republic political system. Also, to realize the distance between these times, and the primary sources who wrote about them, there is considerable chronology. For example, Livy wrote his History of Rome at the court of Augustus at the beginning of the Principate period right around 25BCE, almost 500 years apart! That's a lot of time for legends to evolve and be embellished.

If you want anything clarified let me know as well. Hope this helped!

Source:

Ward, Allen Mason, Fritz M. Heichelheim, and Cedric A. Yeo. A History of the Roman People. Upper Saddle River, NJ: Prentice Hall, 2003. Print.