There does not seem to be an abundance of evidence, although it is the opinion of scholars that King David likely existed.
The evidence we do have:
The Tel Dan Stele
The Tel Dan Stele is written in Aramaic script without vowels which makes translation difficult, allowing for multiple interpretations. Scholars seem to agree that the first part of a phrase refers to house, but disagree if the second part of the phrase is "beloved", "uncle", or "house".
The consensus seems to be that the stele refers to a "House of David":
The inscription, which dates to the ninth century bce, that is to say,
about a century after David was thought to have ruled Israel, includes
the words Beit David ("House" or "Dynasty" of David"). It is the first
near-contemporaneous reference to David ever found. It is not
conclusive; but it does strongly indicate that a king called David
established a dynasty in Israel during the relevant period. - Source
The Mesha Stele
The Mesha Stele is also thought to contain the Name David in two places, which scholars consider a very reasonable translation, however it is not certain. This paper is a series of updates and corrections to an earlier paper, however it still explains the problem in establishing the text on the stele.
I will update with the original paper when I find it.
The City of David
The City of David is a subject of much scholarly debate, of which the Wiki page has a good summary of. There is evidence of pots and sculptures that have been dates to a time that corresponds to what scholars agree would have been King Davids reign.
The evidence we have so far shows only that:
- There may have been a "House of David"
- There was a functioning city and government that corresponds with the time and place King David is thought to have ruled
We don't yet have enough evidence to say for sure that King David existed, or that the claims made about him are true.