There are lots of claims that ancient Israelis were black/brown/white or even east-asian looking? So which one is accurate is there any evidence from Torah or from other biblical resource?
An important source (post-bibical) is the Mishna in Negaim (2:1) which deals with the signs of leprosy and how they may appear different based on a person's skin color:
The bright spot in a German appears as dull white, and the dull white spot in an Ethiopian appears as bright white. Rabbi Ishmael says: the children of Israel (may I be atonement for them!) are like boxwood, neither black nor white but of an intermediate shade.
Based on the description, it appears that the average Jew was darker than a German - (a Northern European) and lighter than an Ethiopian (a Sub-Saharan African). Thus, Jews were neither 'black' nor 'white' but brown like other Mediterranean people.
An additional note - the claims you are hearing are based on a mistaken idea that people were thought of as 'black' or 'white' or 'brown' in ancient times, and the skin color was significant. Racial categories didn't exist before the Middle Ages at earliest. So the skin color was not really focused on except in cases (like Leprosy) where it actually mattered.
Just an addition to what Daniel Kagan writes (very interesting though), Rashi, in his commentary on Bamidbar 12:1 writes:
האשה הכשית THE CUSHITE WOMAN — This tells us that all agreed as to her beauty just as all agree as to the blackness of an Aethopian (cf. Sifrei Bamidbar 99).
Rashi seems to cite the Midrash (Sifrei Bamidbar 99), where it is explained that an Cushite was exceptional in his/her skin:
What is the intent of "Cushite"? Just as a Cushite is exceptional in his skin, so Tzipporah was exceptional in her beauty — more so than all the women. Similarly, (Psalms 7:1) "A Shiggayon of David, which he sang to the L-rd concerning Cush (Saul), a Benjaminite." Now was he a Cushite? (The intent is:) Just as a Cushite is exceptional in his skin
Similary, in Avot d'Rabbi Natan, it is explained that Noah blessed Shem that "made them" dark but comely:
Noah brought his sons and his grandsons, and he blessed them with their (several) settlements, and he gave them as an inheritance all the earth. He especially blessed Shem and his sons, (making them) dark but comely, and he gave them the habitable earth. He blessed Ham and his sons, (making them) dark || like the raven, and he gave them as an inheritance the coast of the sea. He blessed Japheth and his sons, (making) them entirely white, and he gave them for an inheritance the desert and its fields; these (are the inheritances with) which he endowed them.
However, according to the Avot d'Rabbi Natan, Japheth were entirely white and inherited the desert and its fields. It seems to me that this is somewhat problematic, since the Mishnah in Negaim states "The bright spot in a German appears as dull white". The question can be asked if the inhabitants of the Northern Kingdoms (Germany) are from the seed of Japheth?
Here is a list of sources I compiled. This list includes many verses in the Bible that show that Jews were not black, and blackness was as a result of tanning.