Short answer: there is no historical evidence of Moses' life outside of Israelite texts.
Perhaps an appropriate question you didn't ask is should we expect there to be evidence of Moses outside of the texts of ancient Israel? If the stories are false, obviously we would not. If the stories are true, there is good reason to expect other ancient civilizations would not be interested in recording them either because they represent defeat or because they were irrelevant to them.
Among the first category would be the Egyptians who according to the Torah lost a large pool of labor, significant military power, various crops and herds, and every first-born son. It would have been a complete humiliation and a blot on the current pharaoh's name. Assuming such an event were ever recorded, it seems unlikely that the record would be as widely copied as a major victory. (In this sense at least, history really does belong to the victor.) Similarly, the Canaanite peoples who were displaced by the Hebrew people would not be interested or even able to record their own defeat.
In the later category would be the vast majority of the rest of civilization for whom even such a remarkable event as the Exodus would be a minor rumor at best. According to it's own history, Israel was only relevant internationally for the short period of monarchy culminating in the reign of Solomon. Deuteronomy emphasizes the insignificance of the Hebrews at the time:
It was not because you were more in number than any other people that the Lord set his love on you and chose you, for you were the fewest of all peoples, but it is because the Lord loves you and is keeping the oath that he swore to your fathers, that the Lord has brought you out with a mighty hand and redeemed you from the house of slavery, from the hand of Pharaoh king of Egypt.
Therefore, it would be somewhat surprising if we found evidence of the Exodus outside of the Hebrew records. And if the Exodus story were not recorded, then there's no reason to expect the life of Moses to be recorded either.
Tellingly, the earliest extra-Biblical evidence we have that such a people as Israel exists comes as a brief mention of their defeat and extermination (declared prematurely as it turns out) at the hands of Merenptah, an Egyptian pharaoh. The relevant line translated reads:
Israel is wasted, bare of seed,
That is the sort of evidence we might expect outside of Hebrew sources. It's also typical of the records we have from the period: biased and self-serving. (Seriously, read the whole thing.)
The Torah is atypical of the period in that sense—it records both the victories and the follies of the Hebrew people and of Moses. Compared to most ancient rulers, Moses seems surprisingly human. Is that evidence that the Torah dates back to near the end of the events it claims to record? Of course not. But it's the sort of thing that makes dating the early Hebrew texts such an interesting puzzle.
For instance, the first story of Moses' adulthood is an account him murdering an Egyptian:
One day, when Moses had grown up, he went out to his people and looked on their burdens, and he saw an Egyptian beating a Hebrew, one of his people. He looked this way and that, and seeing no one, he struck down the Egyptian and hid him in the sand. When he went out the next day, behold, two Hebrews were struggling together. And he said to the man in the wrong, “Why do you strike your companion?” He answered, “Who made you a prince and a judge over us? Do you mean to kill me as you killed the Egyptian?” Then Moses was afraid, and thought, “Surely the thing is known.” When Pharaoh heard of it, he sought to kill Moses. But Moses fled from Pharaoh and stayed in the land of Midian.
Another oddity of the Torah that doesn't prove anything, but is a puzzle for those holding a late date for it, is that the city of Jerusalem is not mentioned. It isn't until David is said to move the royal residence from Shiloh to Jerusalem that the city is given any importance. The important center for the Hebrew patriarchs according to the Torah seems to be the city of Hebron. Given the reverence that Jerusalem is held in by the majority of Jewish texts, it's surprising that the city remains unnoticed in the Torah. But we are pretty far afield from the original question.
In summary: there is no independent evidence of Moses, there's no reason to expect any, but the evidence in the Torah seems unusual if read as myth and legend.