In Luke 23:38 it is written,
There was a written notice above him, which read:
THIS IS THE KING OF THE JEWS
Why was the notice placed at the crucifixion proclaiming Jesus to be King?
Jesus was hailed as the Messiah - the "Anointed One" (in Greek 'Christos' from which we get Christ). The Messiah was thought to be the one who would restore the kingdom of Israel. (Which, of course, Christians believe he did in the heavenly sense, but not the earthly sense.) You anoint a King, of course, which is where the confusion begins.
Indeed, the "Zealots" of which Simon Zealotes was a member, assumed that Jesus' plan was to be the King of this new newly restored Israel. It was assumed that the Christ would give Israel its land back and be its new ruler. In Jeremiah 23 it says of this Messiah:
“The days are coming,” declares the Lord, “when I will raise up for David[a] a righteous Branch, a King who will reign wisely and do what is just and right in the land. 6 In his days Judah will be saved and Israel will live in safety. This is the name by which he will be called: The Lord Our Righteous Savior.
So, when Pilate is questioning Jesus (in John 18), he inquires along these lines, asking of Jesus if he is a King. To this,
36 Jesus said, “My kingdom is not of this world. If it were, my servants would fight to prevent my arrest by the Jewish leaders. But now my kingdom is from another place.”
37 “You are a king, then!” said Pilate.
Jesus answered, “You say that I am a king. In fact, the reason I was born and came into the world is to testify to the truth. Everyone on the side of truth listens to me.”
The Jews' rejection of Jesus as their king is thus made explicit, when in John 19, Pilate says:
“Shall I crucify your king?” Pilate asked.
“We have no king but Caesar,” the chief priests answered.
Whether or not Jesus was a "King" then, is a matter of interpretation. To be sure, in Revelation, when he returns it is written on his thigh that he is "King of Kings and Lord of Lords", but to lower it to merely earthly place is to denigrate who He really is.
Finally, it should be understood that Pilate was mocking Jesus with the sign. John continues:
19 Pilate had a notice prepared and fastened to the cross. It read: jesus of nazareth, the king of the jews. 20 Many of the Jews read this sign, for the place where Jesus was crucified was near the city, and the sign was written in Aramaic, Latin and Greek. 21 The chief priests of the Jews protested to Pilate, “Do not write ‘The King of the Jews,’ but that this man claimed to be king of the Jews.”
22 Pilate answered, “What I have written, I have written.”
Pilate was contemptuous saying "See, this is what a would-be King of the Jews is". The Chief Priests knew it and tried to change it. But if Pilate's going to mock somebody, he does it fully.
Because had Israel remained independent until Jesus' day, he would have literally been the king of the Jews, according to the two genealogies given at the beginning of Matthew and Luke.
As Wikipedia states,
Both gospels state that Jesus was fathered, not by Joseph, but by God, being born to Mary through a virgin birth. At the same time they claim a direct descent from David, and thus a legal inheritance of the throne.
For someone to be runner up for claiming kingship direct male descent has first claim to it (Numbers 27:6-11). Jesus who claims descent through his mother would be one of the last in line to be king. With David having many sons a direct male descent line is easy, and includes many in a paternal line. Jesus was not in this paternal line but was in the maternal line. He was not eligible because of this.
Numbers 27:6-11 describes how a maternal descent can claim the inheritance only if there is no direct male line. This is not the case in Jesus' situation, hence the Jews and courts did not recognize his authority as king. There's no scriptural authority for his adopted father to pass any kind of lineage, so Jesus was powerless in his claims to be king.
This did not stop Jesus from insisting to be king even though almost every one else rejected it based on the facts of his lineage. It just didn't make sense to accept a lesser blood line when there were plenty that actually fit God's qualifications of straight paternal line.
The notice by the Romans calling him the King of the Jews was a mockery. Because it was well known that he did not qualify to be king and he refused to let it go.