"offering Jesus vinegar to drink." Vinegar is a cheap drink that common people drink. The soldiers were mocking Jesus by giving a cheap drink to someone who claims to be a king.
This was also a way for the soldiers to mock Jesus. AT: "We do not believe you are the King of the Jews, but if you are, prove us wrong and save yourself!"
# a sign over him
"a placard at the top of Jesus' cross that stated"
# This is the King of the Jews
The people who put this sign above Jesus were mocking him. They did not really think he was a king.