Most historians believe that the first king, Romulus, is entirely legendary. His name simply means Roman. The next six kings - Numa Pompilius, Tullus Hostilius, Ancus Marcius, Tarquinius Priscus, Servius Tullius and Tarquinius Superbus - probably existed in some form, but after this disagreements begin.

On one side we have historians who are more forgiving with the literary tradition like Livy and Dionysius of Halicarnassus. They tend to think these kings actually existed and actually did some of the things they are attributed with.

On the other side we have more critical historians. They generally believe that these names might have been real kings, but that the texts describing them are entirely legendary. Some of the most radical think the entire regal period is legendary, and even that Rome was never a kingdom at all.

T.J. Cornell writes:

Traditionally there were seven kings, some of whom are probably historical, at least in the sense that men named Numa Pompilius, Tullus Hostilius, and so on, may indeed have ruled at Rome. But that doesn’t take us very far (and is not certain in any case).

Gary Forsythe claims:

The names of the other six kings [i.e., after Romulus] are likely to be authentic recollections of real people, but it also seems probable that few reliable details were known about their reigns.

Early kings are normally interpreted as mythical or semi-mythical. You can notice certain literary tropes, like how a king with one characteristic is succeeded by a king with the opposite characteristic, which suggests they are not real. These are stock characters, not realistic, well-rounded individuals.

Later kings seem to be more human, more realistic, and may be more historically accurate. Some of the institutions or public buildings attributed to them can be confirmed by independent evidence. However, this does not necessarily prove that they existed as they are portrayed in the texts.

Here is a post I made some time ago answering the question What do we know about the kingdom of Rome? which you might also find interesting.

Sources: T.J. Cornell, The Beginnings of Rome; and Gary Forsythe, A Critical History of Early Rome