The following video is useful background to the post below:
The key portion of the Koran that I want to focus you on is verse 116:
The Koran states:
[5.116] And when Allah will say: O Isa son of Marium! did you say to men, Take me and my mother for two gods besides Allah he will say: Glory be to Thee, it did not befit me that I should say what I had no right to (say); if I had said it, Thou wouldst indeed have known it; Thou knowest what is in my mind, and I do not know what is in Thy mind, surely Thou art the great Knower of the unseen things.
Surah 5:116 (YUS) And behold! Allah will say: "O Jesus the son of Mary! Didst thou say unto men, worship me and my mother as gods in derogation of Allah'?" He will say: "Glory to Thee! never could I say what I had no right (to say). Had I said such a thing, thou wouldst indeed have known it. Thou knowest what is in my heart, though I know not what is in Thine. For Thou knowest in full all that is hidden.
What did Jesus say, though?
63But Jesus held his peace. And the high priest answered and said unto him, I adjure thee by the living God, that thou tell us whether thou be the Christ, the Son of God. 64Jesus saith unto him, Thou hast said: nevertheless I say unto you, Hereafter shall ye see the Son of man sitting on the right hand of power, and coming in the clouds of heaven.
John 5:18 Therefore the Jews sought the more to kill him, because he not only had broken the sabbath, but said also that God was his Father, making himself equal with God.
Jesus acknowledged that he was the Son of God, which equated himself with God.
John 4:23 But the hour cometh, and now is, when the true worshippers shall worship the Father in spirit and in truth: for the Father seeketh such to worship him.
Jesus did not derogate the Father.
John 10:30 I and my Father are one.
But Jesus did claim to be God with the Father.
John 9:35 Jesus heard that they had cast him out; and when he had found him, he said unto him, Dost thou believe on the Son of God?
Indeed, Jesus commanded men to believe on Him.
Hebrews 1:6 And again, when he bringeth in the firstbegotten into the world, he saith, And let all the angels of God worship him.
And God commanded the angels to worship Him.
Therefore, we can see that Jesus did claim divinity. The Jews crucified him for that. The Koran's portrayal of Jesus, therefore, is misleading and incorrect. Jesus is God, but that is no derogation of the Father.
UPDATE: Nick Norelli has some similar thoughts (link).