Most non-Christian scholars will have you believe that Jesus never thought of himself as having personal identity with the Jewish God. First, though skeptical scholars accept enough information to strongly believe in miracles, they tend to challenge the authenticity of many of Jesus words. It just so happens that the words they like to challenge happen to be bold statements of his deity. However, I will argue that it is in fact extremely likely that Jesus indeed thought of himself as God, and very much intended his disciples think so as well.
Modern Scholarship is Not Particularly Concerning
Before we get to the actual evidence, an important point must be made. It would be very logical for a person in 1st century Judaism to not be too forward about the fact that they think they are actually the incarnation of the Jewish God. (Whether or not Jesus actually was forward about it or not). So if the liberal scholarship is in a habit of using a very strict criteria for the sayings of Jesus, it is no surprise at all that they rule out extremely explicit sayings to his deity, since they might not occur very often anyway.
So first off, it makes sense that if he did make such a claim, that he might not necessarily make a lot of noise about it. So if scholars apply ruthless criteria to the text for authenticity, then it shouldn't bother us that they don't come to the conclusion that he claimed he was God, since that is something he may not have explicitly said very much. (Implicitly, it's a whole 'nother ball game!)
All that aside, there is powerful evidence Jesus actually did think of Himself as Israel's God for this main reason:
Why Should the Jews Become Idolaters for No Reason?
The disciples of Jesus and Paul never intended to depart from Judaism, or to stop worshipping Israel's God, or even to stop being monotheists. The apostle Paul, in his letters, is a monotheist. Furthermore, merely being the Jewish Messiah did not automatically mean a person was God. There were a few other people claiming to be the Jewish Messiah in that general time frame, and non of them were worshiped as God, as far as I can tell. (see wikipedia on 1st Century Messiah claimants). Now, in order to say that Jesus didn't want his disciples to think he was God, that means they made it up. But remember...they have no intentions of changing the God they worship. That would mean they would have deliberately risked becoming idolaters for no reason at all. They risk the judgment and the rejection of God himself if they worship other gods. Idolatry is the most serious sin in Judaism. To say the disciples and Paul started worshiping Jesus as God, without him ever saying a word about it, would mean they became idolaters for no reason and without any explanation at all.
The disciples believing that Jesus is God without him saying a word about it, would be akin to 12 fundamentalist Christians deciding that their pastor is God, without him saying anything at all of the sort! Keep in mind, Christians think its the worst evil to worship someone besides Jesus. It's absolutely incredible to assume a monotheist in a particular religion would risk idolatry and rejection from God by worshiping someone who never claimed to be God.
Early Worship of Jesus
In case their is any doubt, the Christians worshiped Jesus as God very early on. In the indisputed letters of Paul, he regularly applies deity to Jesus, within the context of monotheism. The earliest books of the New Testament ascribe deity to Jesus. (1 Thessalonians 4-5 - 50's A.D.) (14) Almost every book of the New Testament paints Jesus as God, or "the Lord" of the Old Testament. Philippians (62 A.D.) (14) quotes an early Christian hymn were Jesus is worshiped as God. The earliest Christian creed "Jesus is Lord," (1 Cor 12:3; Rom 10:9) and the earliest Christian prayer "Come, Lord!" (1 Cor. 16:22; see footnote on Biblegateway) also show Jesus as being worshipped early on. (15) Apostle Peter referred to the day God would visit us when referring to the second coming (1 Peter 65 A.D.). (14) James, the brother of Jesus, refers to "the Lord's coming." (48 A.D.) (14) There was no debate among the early apostles regarding who Jesus was supposed to be.
What Did He Say?
Keep in mind, if we don't use the ruthless criteria applied by the scholars for the authenticity of Jesus words, we can very quickly get to the fact that he claimed to be God. These are scattered all over the Gospels, but especially John, most notably. Jesus said "Anyone who has seen me has seen the Father" (John 14:9) "I and the Father are one." (John 10:33) "Before abraham was born, I am" (John 8:58) "that all may honor the Son just as they honor the Father" (John 5:23) Jesus accepted worship (John 9:38). Jesus claims to have glory with the Father before the world began. (John 17:5). To list a few examples just in John, though there are many elsewhere as well.