The man said, 'Who made you ruler and judge over us?
I will write more about the possible identities of the Pharaoh and his daughter during these events at a later time. For now, let me say that Moses had become quite a part of Pharaoh's family. Acts 7:22 (NIV) sums up these years in the following way:
Other writers also attest to Moses deeds. Philo of Alexandria (a Hellenistic Jewish philosopher) and Josephus (a Jewish historian with a priestly and royal background) both speak about Moses in their writings. Their accounts of Moses' life and actions are quite interesting, and though they do not have the authority of Scripture, they may still provide some interesting insights and details.
Philo provides details about how Moses came to be accepted in Pharaoh's household. [On the Life of Moses, Part 1, Sections IV and V] He speaks of how Pharaoh's daughter pretended to be pregnant and managed to claim Moses as her own child. At first I thought that it was unlikely that she would be able to pull off such a maneuver, but then I considered her position and the fact that as the Pharaoh's beloved daughter, she would not be too closely questioned. It's just in the category of speculation at this point in my mind, but it would make more sense than that Pharaoh would allow her to adopt one of the very Hebrews he was trying to eliminate. Or, perhaps he just allowed his beloved daughter to adopt the child because she wanted it so much. Philo also mentions that the reigning Pharaoh had no sons to continue his dynasty, and that Moses may eventually have been considered an acceptable alternative. At any rate, God allowed Moses to grow up in the household of the Pharaoh and to be educated in all of the learning of the Egyptians. (Acts 7:22)
Josephus goes into some detail about Moses' life as well. In his account, Pharaoh's daughter, who is named as Thermuthis, admits to her father that she received the child from the river. Shortly before that time, Egyptian scribes had warned the king that a child would arise from the Hebrew nation who would be famous and would lead to the downfall of the Egyptian nation, so they immediately urged that Moses should be destroyed. However, Thermuthis would have none of that, and snatched him from their hands. Largely because there were no other likely prospects for the throne, Moses is allowed to live, although he is viewed with suspicion by the Egyptians. Years later, when Egypt runs into trouble with the Ethiopians, Pharaoh orders Moses to act as one of his generals (Pharaoh having assured his daughter that the Egyptians would not harm him). Moses is successful in many engagements, so much so that when he is attacking Saba (a city protected on all sides by rivers and fortifications) an Ethiopian princess (Tharbis) arranges to deliver the city into Moses' hands if he will agree to take her as a wife. Moses agrees and the city is overthrown.
Even at that point, the Egyptians remained suspicious about Moses' intentions, thinking that he will make use of his success to gain power for himself. The Pharaoh himself is jealous of Moses' success and nervous about his own status. [The Writings of Flavius Josephus, Antiquities of the Jews, Book 2, chapters 9-11] So, at the time of the killing of the Egyptian overseer in the Biblical account above, it is not surprising that the Pharaoh or the Egyptian court would have been seeking any opportunity possible to discredit Moses.