When Christ started His ministry, He appointed 12 apostles. When Judas betrayed Jesus, did he kill himself before or after Jesus died on the cross? After Judas died, they appointed another to take His place. Why?

It is thought that Judas killed himself before Jesus actually died on the cross. According to Matthew’s gospel, Early in the morning, all the chief priests and the elders of the people came to the decision to put Jesus to death. They bound him, led him away and handed him over to Pilate, the governor. When Judas, who had betrayed him, saw that Jesus was condemned, he was seized with remorse and returned the thirty silver coins to the chief priests and the elders. “I have sinned,” he said, “for I have betrayed innocent blood.” “What is that to us?” they replied. “That’s your responsibility.” So Judas threw the money into the temple and left. Then he went away and hanged himself—Matthew 27:1-5. A few verses later, Matthew tells us, Meanwhile, Jesus stood before the governor—Matthew 27:11a. So if we understand the timeline correctly, Jesus is still “in process” with all those authorities dealing with him while Judas is already dead. Matthew is the only one of the gospel writers who details Judas’ death, but Luke, although he doesn’t deal with it in his gospel, does take up the subject in the book of Acts, telling us, In those days Peter stood up among the believers (a group numbering about a hundred and twenty) and said, “Brothers, the Scripture had to be fulfilled which the Holy Spirit spoke long ago through the mouth of David concerning Judas, who served as guide for those who arrested Jesus—he was one of our number and shared in this ministry.” (With the reward he got for his wickedness, Judas bought a field; there he fell headlong, his body burst open and all his intestines spilled out. Everyone in Jerusalem heard about this, so they called that field in their language Akeldama, that is, Field of Blood.)—Acts 1:15-19. Apparently after he hanged himself, either the rope or the tree limb holding him broke and he fell, his body breaking open as a result.
As far as replacing Judas is concerned, as Peter continued talking, he quoted from Psalm 109:8 which says, [M]ay another take his place of leadership—David, Psalm 109:8b. Peter appears to have been led by the Holy Spirit as found in the Scriptures to determine that it was indeed appropriate and necessary for someone to take Judas’ place. There were a couple of requirements: (1) It had to be someone who had participated in Jesus’ earthly ministry, and (2) it had to be someone who had seen the resurrected Christ, incredibly important since the resurrection was a key to apostolic teaching. Luke tells us what happened next: So they proposed two men: Joseph called Barsabbas (also known as Justus) and Matthias. Then they prayed, “Lord, you know everyone’s heart. Show us which of these two men you have chosen to take over this apostolic ministry, which Judas left to go where he belongs.” Then they cast lots, and the lot fell to Matthias; so he was added to the eleven apostles—Acts 1:23-26. They used “lots,” a tried and true Old Testament method of seeing God’s will. This is, incidentally, the last biblical reference of the use of lots. The coming of the Holy Spirit would make lots no longer necessary. But ultimately, God made the choice. Interestingly, Eusebius, an early church historian, suggests that Matthias might have been among the 70 sent out by Christ as recorded in Luke 10:1: After this the Lord appointed seventy-two others and sent them two by two ahead of him to every town and place where he was about to go—Luke 10:1.