What did Jesus think he was? What does the biblical Jesus (as opposed to the historical Jesus) think he is?


a rabbi
the son of a carpenter
the child born in the house of David
the king of the Jews, about to be anointed with oil, to save Jerusalem from all oppression and be the light of the world
a pious Jewish man well-versed in the Hebrew scriptures at the time and carefully cites some sources from the Hebrew scriptures in order to serve his purpose and make himself seem credible, genuine, and sincere to his community
a traveler who wants anybody who has ears to hear to listen to him
a man holding a grudge against the Pharisees, even though one of the Pharisees shows hospitality by inviting him to his house
a prophet that foretells what shall happen to the Son of Man, which turns out to be him
his disciples seem to treat him like a teacher, a rabbi, and sometimes refer to him as "lord" (whether or not "Lord" is the correct translated term is another story)
complex relationship with his "Father"


Compared to modern Rabbinical Jewish mashiach, it is implied that Jesus presumably thought that he was the Jewish messiah, the King of the Jews. Wearing a crown of thorns and being labelled "King of the Jews" at his crucifixion and resurrection. Plus, some say that his resurrection really did happen in history. His resurrection seems to be most significant, because without it, he would be a dead messiah, and that means he would be no messiah.
The common Christian answer goes straight to the words Jesus spoke in John 8:58 


  Jesus said to them, "Most assuredly, I say to you, before Abraham was,
  I AM."


This is understood now, and was understood at the time, as Jesus claiming to be God, referring to the the words God spoke to Moses from the burning bush.

This is covered in much greater detail at Let Us Reason Ministries.

Excerpt:


  Who do you say Jesus is? Jesus asked this same question to his
  disciples about what others thought of him and then asked what they
  thought of him.
  
  Jn.5:37: "And the Father himself, who sent me, has borne witness of
  me. You have neither heard his voice at any time, nor seen his shape.
  ‘Jesus gives the source of his commission, which is from the Father
  personally. It is the Fathers voice and form they have not seen, yet
  Christ has.
  
  Christ who is called the exact image of the invisible Father is the
  voice that the people heard. He then says that they search the
  Scriptures in them you think you have eternal life but they testify of
  me."(v.39) The Son is said to be the eternal life with the Father. Are
  we to believe the Scriptures testify of only a human being and not God
  himself? In the end of the discourse Jesus says in vs.46-47 "If you
  believed Moses you would believe Me; for he wrote about me. But if you
  don’t believe his writings, how will you believe my words?"
  
  When did Moses write of him? Deut.18:15-19: "The LORD your God will
  raise up for you a Prophet like me from your midst, from your
  brethren. Him you shall hear, "according to all you desired of the
  LORD your God in Horeb in the day of the assembly, saying, 'Let me not
  hear again the voice of the LORD my God, nor let me see this great
  fire anymore, lest I die.' "And the LORD said to me: 'What they have
  spoken is good. 'I will raise up for them a Prophet like you from
  among their brethren, and will put My words in His mouth, and He shall
  speak to them all that I command Him. 'And it shall be that whoever
  will not hear My words, which He speaks in My name, I will require it
  of him." Jesus claims to be the prophet Moses spoke of that should
  listen to. Notice that it says they did not want to hear the voice of
  the Lord anymore or see his glory in Horeb. Then God says he will put
  his words in a future prophets mouth if they do not listen to his
  words, God will require it of him." This very thing Jesus said of
  himself in Jn.8:24 "Therefore I said to you that you will die in your
  sins; for if you do not believe that I am (He), you will die in your
  sins."


This is also famously covered in C.S. Lewis's Trilemma


  "I am trying here to prevent anyone saying the really foolish thing
  that people often say about Him: I’m ready to accept Jesus as a great
  moral teacher, but I don’t accept his claim to be God. That is the one
  thing we must not say. A man who was merely a man and said the sort of
  things Jesus said would not be a great moral teacher. He would either
  be a lunatic — on the level with the man who says he is a poached egg
  — or else he would be the Devil of Hell. You must make your choice.
  Either this man was, and is, the Son of God, or else a madman or
  something worse. You can shut him up for a fool, you can spit at him
  and kill him as a demon or you can fall at his feet and call him Lord
  and God, but let us not come with any patronising nonsense about his
  being a great human teacher. He has not left that open to us. He did
  not intend to. ... Now it seems to me obvious that He was neither a
  lunatic nor a fiend: and consequently, however strange or terrifying
  or unlikely it may seem, I have to accept the view that He was and is
  God."