Why did Peter keep preaching to Israel after Jesus had been killed?
Peter kept preaching to Israel because the covenant promises required Israel to believe first; he hoped against hope that they would wake up and recognize the One they killed as the Promised Messiah. Their act was done in ignorance, leaving room for forgiveness.
Les explains that Jesus had presented Himself to the nation of Israel for three years as the Messiah and Promised One according to the covenants. Since it had to go to the Jew first and then to the Gentile, Peter kept coming back, hoping against hope that Israel would wake up and realize the One they killed was the Promised Messiah. Peter would not address Gentiles as brethren at this point because they had no idea they were to have anything to do with Gentiles. He told Israel that through ignorance they had killed the Messiah, as did their rulers. Les notes that Christ from the cross said, "Father forgive them," and Stephen in chapter 7 said basically the same thing: forgive them, for they know not what they do.