set your mind on things above, not on things on the earth(Colossians 3:2)
Put on the new man who is renewed in knowledge according to the image of Him (Colossians 3:10)
To understand the peace of Christ, we must go back to the beginning of the chapter. Brethren had disputes about food, drinks, and feast days, which Paul points out were "shadows of things to come," but the actual fulfillment of those things is in Christ (Colossians 2:16-17). So, beginning in chapter three, Paul opens up about the spiritual life that one has in Christ. He tells the brethren not to be so carnally minded, "set your mind on things above, not on things on the earth" (Colossians 3:2).
He then moves into the true purpose and the key to understanding Christ's true peace. In verse five, he begins to list things that wage war against the spirit, which ultimately never bring peace but sadness, misery, disappointment, turmoil, sin, and spiritual death. Fornication, uncleanness, passions, evil desire, and covetousness kill the spirit through war. The antithesis of peace is war. Therefore, to have the peace of Christ, who never fell victim to any of these things, one must put these things that war against the soul to death. But if that was not enough, he gives another list in verses eight and nine. Anger, wrath, malice, blasphemy, filthy language, and lying kill the spirit.
Now, one may object and say, "Well, Christ never committed such carnal things, yet He did not live peaceably with all men; I mean, He was killed by His enemies." That is where we find the breakdown and misapplication of what the peace of Christ is. Christ indeed had enemies, but despite having them, He did allow Himself to act as humanity did and still does. He could live His life with the same peace He offers His followers.
No one can have the peace of Christ unless they have, first, been raised with Christ (Colossians 3:1), and second, "put on the new man who is renewed in knowledge according to the image of Him who created him" (Colossians 3:10). Then, we see Paul saying it doesn't matter if you are "Jew or Gentile, circumcised nor uncircumcised, barbarian, Scythian, slave nor free," we all struggle with these things because we are human. And the only way to have true peace is to find true identity in Christ Jesus (Colossians 3:11). And if we are in Christ, then we must forgive one another and love one another, which is the "bond of perfection" (Colossians 3:13-14). This brings us to Colossians 3:15, when we can stop acting like humans with carnal minds; then we can become followers of Christ, people who walk by the Spirit, thus, bringing the peace of Christ to all our lives.