Concerning Colossians 1:24, Paul seems to be saying that the suffering of Christ on our behalf was not complete and so Paul had to finish it. How does that work?

Paul wrote, Now I rejoice in what was suffered for you, and I fill up in my flesh what is still lacking in regard to Christ’s afflictions, for the sake of his body, which is the church—Paul, Colossians 1:24. This is a good example of needing to consider these words in the “context” of all that Paul is saying at this point in his letter. The thought really begins back in verse 19, where Paul says, For God was pleased to have all his [God’s] fullness dwell in him [Jesus Christ], and through him [Jesus Christ] to reconcile to himself [God] all things, whether things on earth or things in heaven, by making peace through his [Jesus Christ’s] blood, shed on the cross. Once you were alienated from God and were enemies in your minds because of your evil behavior. But now he [God] has reconciled you by Christ’s physical body through death to present you holy in his [God’s] sight, without blemish and free from accusation—if you continue in your faith, established and firm, not moved from the hope held out in the gospel. This is the gospel that you heard and that has been proclaimed to every creature under heaven, and of which I, Paul, have become a servant—Paul, Colossians 1:19-23. Now, what Paul is saying in verse 24 is that he is rejoicing in the sufferings he (Paul) has endured on behalf of the Colossians (and, in fact, all the churches of Jesus Christ) because he knows his sufferings are helping to build up Christ’s church. We need to remember that, at the time of these words, Paul is imprisoned in Rome. Paul was in a sense experiencing the persecution that had begun with Christ’s enemies persecuting Him. The persecution inflicted on Christ did not occur because of who He was as a man, but rather on the things He insisted on teaching and preaching—His message. Christ’s death on the cross did not end the persecution because the message did not end; it simply continued on through Peter, Paul, and the rest of the apostles, and then on through the growing followers of the gospel. Paul is not saying that Christ’s sufferings were not complete in the sense of inadequate; it is saying that Christ’s sufferings were not complete in the sense that the world was not yet finished with what it wanted to do to those who claimed His name. Christ had warned His followers when He was here: Then you will be handed over to be persecuted and put to death, and you will be hated by all nations because of me—Jesus Christ, Matthew 24:9. The sufferings He endured were simply indicative of what His followers would also endure: But before all this, they will lay hands on you and persecute you. They will deliver you to synagogues and prisons, and you will be brought before kings and governors, and all on account of my name. This will result in your bearing witness to them. But make up your mind not to worry beforehand how you will defend yourselves. For I will give you words and wisdom that none of your adversaries will be able to resist or contradict. You will be betrayed even by parents, brothers, relatives and friends, and they will put some of you to death. All men will hate you because of me—Jesus Christ, Luke 21:12-17.
Yet how did Christ’s followers respond to the persecutions that they faced? When Peter and John were arrested for preaching the gospel, the Sanhedrin called the apostles in and had them flogged. Then they ordered them not to speak in the name of Jesus, and let them go. The apostles left the Sanhedrin, rejoicing because they had been counted worthy of suffering disgrace for the Name—Acts 5:40-41. In the town of Philippi, missionaries Paul and Silas were arrested, flogged, and thrown into prison with their feet held in stocks. Yet Luke tells us that [a]bout midnight Paul and Silas were praying and singing hymns to God, and the other prisoners were listening to them—Acts 16:25. As Paul reminds his readers in his second letter to the church at Corinth, For we do not preach ourselves, but Jesus Christ as Lord, and ourselves as your servants for Jesus’ sake. For God, who said, “LET LIGHT SHINE OUT OF DARKNESS,” made his light shine in our hearts to give us the light of the knowledge of the glory of God in the face of Christ. But we have this treasure in jars of clay to show that this all-surpassing power is from God and not from us. We are hard pressed on every side, but not crushed; perplexed, but not in despair; persecuted, but not abandoned; struck down, but not destroyed. We always carry around in our body the death of Jesus, so that the life of Jesus may also be revealed in our body. For we who are alive are always being given over to death for Jesus’ sake, so that his life may be revealed in our mortal body. So then, death is at work in us, but life is at work in you—Paul, II Corinthians 4:5-12. As he pointed out to them regarding his life as a minister of the gospel, [A]s servants of God we commend ourselves in every way: in great endurance; in troubles, hardships and distresses; in beatings, imprisonments and riots; in hard work, sleepless nights and hunger; in purity, understanding, patience and kindness; in the Holy Spirit and in sincere love; in truthful speech and in the power of God; with weapons of righteousness in the right hand and in the left; through glory and dishonor, bad report and good report; genuine, yet regarded as imposters; known, yet regarded as unknown; dying, and yet we live on; beaten, and yet not killed; sorrowful, yet always rejoicing; poor, yet making many rich; having nothing, and yet possessing everything—Paul, II Corinthians 6:4-10. His life was littered with difficulties; as he himself said, I have worked much harder, been in prison more frequently, been flogged more severely, and been exposed to death again and again. Five times I received from the Jews the forty lashes minus one. Three times I was beaten with rods, once I was stoned, three times I was shipwrecked, I spent a night and a day in the open sea, I have been constantly on the move. I have been in danger from rivers, in danger from bandits, in danger from my own people, in danger from Gentiles; in danger in the city, in danger in the country, in danger at sea; and in danger from false brothers. I have labored and toiled and have often gone without sleep; I have known hunger and thirst and have often gone without food; I have been cold and naked—Paul, II Corinthians 11:23b-27. Yet, with all this difficulty in his life as a direct result of being a messenger of the gospel, what did he say? But whatever was to my profit I now consider loss for the sake of Christ. What is more, I consider everything a loss compared to the surpassing greatness of knowing Christ Jesus my Lord, for whose sake I have lost all things. I consider them rubbish, that I may gain Christ and be found in him, not having a righteousness of my own that comes from the law, but that which is through faith in Christ—the righteousness that comes from God and is by faith. I want to know Christ and the power of his resurrection and the fellowship of sharing in his sufferings, becoming like him in his death—Paul, Philippians 3:7-10. He understood that the persecutions directed towards Jesus were continued on in him—and he was thankful, even rejoicing, in this. He knew that his moments of pain and misery were building up the church—encouraging it to reflect Jesus no matter what. That’s why he could literally celebrate, [W]e know that the one who raised the Lord Jesus from the dead will also raise us with Jesus and present us with you to himself. All this is for your benefit, so that the grace that is reaching more and more people may cause thanksgiving to overflow to the glory of God. Therefore we do not lose heart. Though outwardly we are wasting away, yet inwardly we are being renewed day by day. For our light and momentary troubles are achieving for us an eternal glory that far outweighs them all. So we fix our eyes not on what is seen, but on what is unseen, since what is seen is temporary, but what is unseen is eternal—Paul, II Corinthians 4:14-18.