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... to have equality with God the Father.
Only God is capable of certain acts. Those of Jesus' generation knew this well and many believed HE was worthy of death for claiming those capabilities.

But He answered them, "My Father is working until now, and I Myself am working." For this cause therefore the Jews were seeking all the more to kill Him, because He not only was breaking the Sabbath, but also was calling God His own Father, making Himself equal with God.(John 5:17,18)

"For just as the Father raises the dead and gives them life, even so the Son also gives life to whom He wishes. For not even the Father judges anyone, but He has given all judgment to the Son, in order that all may honor the Son, even as they honor the Father. He who does not honor the Son does not honor the Father who sent Him." (John 5:21-23)

Who else but God above can raise the dead and give life, both physically and spiritually? Jesus claimed to be capable of the same abilities as the Father and, therefore, deserving of the same honor. (see also John 8:10)

... to be self-existent.
Who but God alone is self-existent?
"For just as the Father has life in Himself, even so He gave to the Son also to have life in Himself" (John 5:26) Jesus actually claimed to have life in Himself just as God the Father does! Obviously, Jesus is claiming to be much more than just a good teacher or religious leader.

... that the Scriptures speak about Himself.
The Jews constantly sought to find eternal life through the Old Testament Scriptures.

"You search the Scriptures, because you think that in them you have eternal life; and it is these that bear witness of Me; and you are unwilling to come to Me, that you may have life." (John 5:39,40)

What arrogance and presumption a mere man would display in making such a claim. And yet Jesus, as God, had every right to make such an assertion.