Did Jesus claim to be God?

Yes, aside from his disciple's belief that he was God, Jesus made several direct claims that he was God. In many instances, you must take into consideration the Old Testament claims of God and the beliefs of the Jews in Jesus' day.

        "I tell you the truth," Jesus answered, "before Abraham was born, I am!" (John 8:58)

Among the claims of being God's only begotten son by both Jesus and God himself (Matthew 3:17, 11:27, 16:13-17, 17:5, 22:41-45, 26:63-64, 28:18-19, Mark 14:62, Luke 1:35, 9:35, 10:22, 22:67-71, John 1:32-34, 1:49-50, 3:16-18, 5:17-22, 5:25-26, 5:41-43, 6:40, 8:54-56, 10:34-38, 11:4, 11:25-27, 14:6-7, 14:8-14, 14:20, 15:8-10, 16:27-28, 17:1-5, 17:20-25, 19:7, 19:17), Jesus also explained his intricate interrelationship with God (John 5:19-22, 5:25-26, 14:9-14, 17:20-25) and made references to being the Almighty in various settings where those around him definitely got the message: authority to forgive sins, Mark 2:5-12; Son of Man is Lord of the Sabbath, Luke 6:5; authority to judge the living and the dead, John 5:27. It wasn't enough that he claimed to be the Son of God, but the Jews actually considered this a claim of equality with God (John 5:17-18, 19:7) and one that was punishable by death.

        "I and the Father are one." Again the Jews picked up stones to stone him, but Jesus said to them, "I have shown you many great miracles from the Father. For which one of these do you stone me?" "We are not stoning you for any of these," replied the Jews, "but for blasphemy, because you, a mere man, claim to be God." (John 10:30-33)
        "All I have is yours, and all you have is mine... Holy Father, protect them by the power of your name - the name you gave me - so that they may be one as we are one." (John 17:10-11)

Jesus often made the reply "I am," which was not permitted in the Jewish culture of his day since this was the official name of God and the claim by God of God's very existence (Exodus 3:14). Aside from the Apostle Paul (1 Corinthians 15:10--"But by the grace of God I am what I am..."), the writers of the scriptures have been careful not to use this phrase as a single statement, especially since it was an utterance considered punishable by death in Jewish culture.

        Moses said to God, "Suppose I go to the Israelites and say to them, 'The God of your fathers has sent me to you,' and they ask me, 'What is his name?' Then what shall I tell them?"
        God said to Moses, "I AM WHO I AM. This is what you are to say to the Israelites: 'I AM has sent me to you.' "
(Exodus 3:13-14)

        Again the high priest asked him, "Are you the Christ, the Son of the Blessed One?" "I am," said Jesus... The high priest tore his clothes. "Why do we need any more witnesses?" he asked. "You have heard the blasphemy. What do you think?" They all condemned him as worthy of death. (Mark 14:62-63, Luke 22:70)
        "I tell you the truth," Jesus answered, "before Abraham was born, I am!" At this, they picked up stones to stone him... (John 8:58)
        "Who is it you want?" "Jesus of Nazareth," they replied. "I am he," Jesus said. When Jesus said, "I am he," they drew back and fell to the ground. (John 18:4-6)

Jesus also made the claim of being the first and the last, the beginning and the end, another sacred claim held only by God (Isaiah 44:6, 48:12, Revelation 1:8).

        "This is what the Lord says-- Israel's King and Redeemer, the Lord Almighty: I am the first and I am the last; apart from me there is no God. Who then is like me? Let him proclaim it. Let him declare and lay out before me what has happened since I established my ancient people, and what is yet to come -- yes, let him foretell what will come." (Isaiah 44:6-7)
        He who was seated on the throne said, "I am making everything new!" Then he said, "Write this down, for these words are trustworthy and true." He said to me, "It is done. I am the Alpha and the Omega, the Beginning and the End." (Revelation 21:5-6)
        "Behold, I am coming soon! My reward is with me, and I will give to everyone according to what he has done. I am the Alpha and the Omega, the First and the Last, the Beginning and the End... I, Jesus, have sent my angel to give you this testimony for the churches." (Revelation 22:12-16)

And Jesus said that if they saw him, they've seen God, which wasn't just a reference to quality and appearance, but of Godhead incarnate.

        Jesus answered, "I am the way and the truth and the life. No one comes to the Father except through me. If you really knew me, you would know my Father as well. From now on, you do know him and have seen him." (John 14:6-7)