I would like a chronological list of the people the Jewish and
Christian scriptures (aka old and new testament) consider BOTH
prophets AND kings with a reference to the scripture that makes this
claim for each person.

The definitions may need a little work - for instance an Egyptian
Pharaoh and Nebuchadnezzar both had prophetic dreams but were unable
to interpret them unaided.  This is not normally considered being a
prophet.

Abimelech in Genesis had a dream from God.
And Moses wouldn't really count as a king.

Definite prophet-kings would be:
Saul - 1 Sam 10:11 - 1 Sam 11:15
David - Luke 20:42 - 2 Sam 8:10
Completeness not guaranteed...

Then the New Testament has a major character who is called a prophet
and a king (and other things).

This is why I included the line "with a reference to the scripture
that makes this claim for each person." Because the scriptures never
claim that Moses is a king you are right, he wouldn't count. Also
because the scriptures never claim Pharaoh & Nebuchadnezzar are
prophets they would not count. Daniel however would count as a prophet
("Daniel the prophet" - Matthew 24:15)  but not as a king.

I'm look for a list of those who the scriptures (Jewish &/or
Christian) claim are both.