Try T. J. Cornell, The Beginnings of Rome: Italy and Rome from the Bronze Age to the Punic Wars (c.1000-264 BC). If you’re looking for a verifiable account of the kings, you’ll be disappointed, since these stories are only known from myth and legend. Livy, books 1-5, is the main ancient source for this period, and he’s still writing 5-7 centuries after the (alleged) fact. That said, Livy is great and can tell you a lot about what Romans believed about their past.

I particularly enjoyed Religions of Rome: Volume 1: A History by John North, Mary Beard, and Simon Price. It's not an overview — it specifically looks at the religion of Rome during this time, but it's extremely accessible for an academic text and absolutely fascinating, and of course Roman religion isn't something that happens outside the context of politics and common life, so it has a ton of great information. It starts out looking at the early cults during the period of monarchy, then goes on to the republic and empire, so it's not exclusive to the Kingdom of Rome and early republic, but it definitely doesn't just gloss over it.

We've just completed a major revision of the Askhistorians ancient Rome book list, including a whole section on the Regal and Early Republic. Do please check it out!