Why did Jesus come to earth? Asking different Christians will likely give you different answers. Some will say that he came to forgive sins, or to show us God's character. These are both true - but they are not the chief reason. Jesus came to establish his kingdom, and the forgiveness of sins and modeling the character of God are accessories to that main purpose.
“From that time Jesus began to preach, saying, "Repent, for the kingdom of heaven is at hand."”
Matthew 4:17 ESV
http://bible.com/59/mat.4.17.esv
This verse if from the beginning of Jesus's ministry. He is calling on people to repent (a topic for a later study) in preparation for the kingdom of heaven. The Jews of the time would have understood the references and the echoing of John the Baptist's message. They may have even connected the proclamation with the messianic expectations of Isaiah.
“Now when Jesus came into the district of Caesarea Philippi, he asked his disciples, "Who do people say that the Son of Man is?" And they said, "Some say John the Baptist, others say Elijah, and others Jeremiah or one of the prophets." He said to them, "But who do you say that I am?" Simon Peter replied, "You are the Christ, the Son of the living God." And Jesus answered him, "Blessed are you, Simon Bar-Jonah! For flesh and blood has not revealed this to you, but my Father who is in heaven. And I tell you, you are Peter, and on this rock I will build my church, and the gates of hell shall not prevail against it.
Truly, I say to you, there are some standing here who will not taste death until they see the Son of Man coming in his kingdom."”
Matthew 16:13-18, 28 ESV
http://bible.com/59/mat.16.13-18,28.esv
In the above verse Jesus confirms his status as the Christ. He immediately goes into how he will establish his church on the truth of his identity, and that some of the disciples would not die until the kingdom was established. There is disagreement on if Jesus it talking about complete death or merely earthly death. We know that all of those that Jesus was addressing died earthly deaths, so the question is if the coming of the kingdom references a relationship with Jesus now, or if it is referencing Jesus's second coming and judgement day. I tend to think that the first is more likely, but the second is possible since we know that Jesus promises eternal life to his followers (so the death that he is referencing in this case may be spiritual). Nevertheless, there is a clear emphasis on Jesus's Kingdom.
“So Pilate entered his headquarters again and called Jesus and said to him, "Are you the King of the Jews?" Jesus answered, "Do you say this of your own accord, or did others say it to you about me?" Pilate answered, "Am I a Jew? Your own nation and the chief priests have delivered you over to me. What have you done?" Jesus answered, "My kingdom is not of this world. If my kingdom were of this world, my servants would have been fighting, that I might not be delivered over to the Jews. But my kingdom is not from the world." Then Pilate said to him, "So you are a king?" Jesus answered, "You say that I am a king. For this purpose I was born and for this purpose I have come into the world—to bear witness to the truth. Everyone who is of the truth listens to my voice." Pilate said to him, "What is truth?" After he had said this, he went back outside to the Jews and told them, "I find no guilt in him.”
John 18:33-38 ESV
http://bible.com/59/jhn.18.33-38.esv
Jesus tells Pilate that his kingdom is not of this world and that he came to bear witness to the truth. Jesus's kingdom is truth and those who accepts his message is of the truth. Too often we may try to establish Jesus's kingdom here on earth built by human hands. Theocracy is not what Jesus is calling us to - he already has a kingdom that we are members of through our death in baptism. But it is not of this world, so we cannot strive to be of this world - otherwise we are leaving Jesus's kingdom for a lesser earthly kingdom of our own making or just an earthly government. Unlike Islam or Judaism, Christianity calls us to belong to a heavenly kingdom headed by Jesus rather than trying to establish a theocratic government headed by man.
“But seek first the kingdom of God and his righteousness, and all these things will be added to you.”
Matthew 6:33 ESV
http://bible.com/59/mat.6.33.esv
As Christians, we should be striving to focus on God's kingdom rather than the anxieties or needs of this world.
“At that time the disciples came to Jesus, saying, "Who is the greatest in the kingdom of heaven?" And calling to him a child, he put him in the midst of them and said, "Truly, I say to you, unless you turn and become like children, you will never enter the kingdom of heaven. Whoever humbles himself like this child is the greatest in the kingdom of heaven. "Whoever receives one such child in my name receives me, but whoever causes one of these little ones who believe in me to sin, it would be better for him to have a great millstone fastened around his neck and to be drowned in the depth of the sea. "Woe to the world for temptations to sin! For it is necessary that temptations come, but woe to the one by whom the temptation comes! And if your hand or your foot causes you to sin, cut it off and throw it away. It is better for you to enter life crippled or lame than with two hands or two feet to be thrown into the eternal fire. And if your eye causes you to sin, tear it out and throw it away. It is better for you to enter life with one eye than with two eyes to be thrown into the hell of fire.”
Matthew 18:1-9 ESV
http://bible.com/59/mat.18.1-9.esv
A key characteristic of being a part of the kingdom of heaven is humility like a child. Jesus also highlights how sin distances us from the kingdom of God.
“Now there was a man of the Pharisees named Nicodemus, a ruler of the Jews. This man came to Jesus by night and said to him, "Rabbi, we know that you are a teacher come from God, for no one can do these signs that you do unless God is with him." Jesus answered him, "Truly, truly, I say to you, unless one is born again he cannot see the kingdom of God." Nicodemus said to him, "How can a man be born when he is old? Can he enter a second time into his mother's womb and be born?" Jesus answered, "Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God.”
John 3:1-5 ESV
http://bible.com/59/jhn.3.1-5.esv
Entering the kingdom of God requires a second birth of water and the spirit. In baptism, we die to the world and have a new birth in Christ.
Further reading Matthew ch 5 and ch 12 about the Kingdom of heaven.