ANSWER: While the Bible does not state all of reasons God used to justify wanting to kill Moses, we can take what information is available and derive possible explanations. Although it may seem outwardly there was nothing wrong, to God nothing is hidden!
The issue of punishing Moses by killing him surrounds the circumcision of a son. It seems it was not done in a timely and proper manner, on the eighth day after birth, as God commanded.
The Bible states that while Moses was in Midian God told him to go back to Egypt (Exodus 4:19). He soon travels to Egypt with his wife and sons. Exodus states that during the journey "the Lord met Moses and tried to kill him." It is then that Zipporah, his wife, did something seemingly odd.
And it came to pass by the way, in the inn, that the Lord met him and sought to kill him. And Zipporah took a sharp stone, and cut off the foreskin of her son, and threw it at his feet, and said, "Surely a bloody husband you are to me." (Exodus 4:24 - 25, HBFV)
Zipporah's rather assertive act saved her husband from certain death (verse 26).
Because only one of his sons is mentioned here - not having been circumcised, we must assume the other was! Evidence suggests that Moses did circumcise his firstborn according to the command of God. It was what he did not do for his second son that could have gotten him killed!
Was Zipporah (not being an Israelite) so appalled at the bloody act that she gave her husband a hard time over it - such that he didn't want to do it again? God knew about the situation and gave him time to repent. Moses may have hoped the problem would be forgotten and go away.
Circumcision on the eighth day after birth was part of the covenant between Abraham and God (Genesis 17:10 - 14). This act became part of the agreement between God and Israel. Every male child had to be circumcised or else 'that person shall be cut off from his people' (Genesis 17:14).
Not only was Moses the head of his household, but God had picked him to be head of the whole nation of Israel - he was now going to make a decision - circumcise the boy - or be killed! Him and his wife must have discussed the matter and she knew he was dying before her eyes, making her quick to circumcise the boy! But she was still upset over having to circumcise her son. There is more that is not recorded that took place, but is alluded to in Exodus 18.
"Then Jethro . . . after he had sent her back, with her two sons, of whom the name of one was Gershom . . . the other was Eliezer" (Exodus 18:2 - 4).
Why had Moses sent his wife and his sons back? It is possible that because his son was most likely a young boy, not a baby or small child, circumcision would have made him unable to travel without great pain. The whole ordeal with his wife over the circumcision likely caused him to send his family back to his father-in-law's house.
After the drama of the circumcision Aaron, the brother of Moses, was commanded by God to meet him in the wilderness (Exodus 4:27). Zipporah almost certainly stayed at the Inn with her son until he was able to travel. This likely influenced the decision of Moses to send his family back to Jethro's house.
Discerning the scriptures and how Zipporah was a woman who spoke her mind, and most likely not have let up on her husband all the while they were in Egypt, it made some sense (from the point of view of Moses) to send her back.
Zipporah took the foreskin of the circumcised boy and threw it at the feet of Moses along with mouthing her disgust for doing it! She had successfully save her husband from being killed. God, however, no more mentions Zipporah after Exodus 18. Additionally, her sons are not mentioned with any degree of importance anywhere else in the Bible.