When Jesus was asked by the Roman Governor, "Are you the king of the Jews?", his reply was "Thou sayest" (A polite Aramaic way of saying "yes"). Pilate referred to him as the king of the Jews a few minutes later (Mark 15:2, Mark 15:9). The magi came looking for "the king of the Jews" (Matthew 2:2).
What is Jesus' authentication as king?
God had promised David, the greatest of Israel's ancient kings, that his descendant would reign on his throne forever (2 Samuel 7:12-16). Four hundred years after David and six hundred years before Jesus, there lived an evil king called Coniah who was also a descendant of David and Solomon. God cursed him and said that none of his descendants would ever be king over Israel (Jeremiah 22:24-30)! Jesus' foster father Joseph was a descendant of David via the royal line through Solomon and Coniah (see Matthew 1). Joseph is thus disqualified to be king over Israel, but Jesus had a virgin birth, so the same disqualification does not apply to him! His mother Mary was a descendant of David through Nathan (see Luke 3:23 onwards). Thus, Jesus is legally and biologically the heir to the throne of David.
Note that these genealogical records were well known at the time of Jesus' birth. In the census, his parents were registered at Bethlehem, David's home town, because it was common knowledge that they were his descendants (Luke 2:1-5). These records were destroyed when the Romans crushed the Jewish rebellion of AD 70.
But Israel rejected her king (John 19:15). However, since the kingship of Jesus is an eternal purpose of God (Psalm 2:6-7), it cannot be annulled by human rebellion. Thus, Jesus will be king over Israel in the future. The Bible tells us how this will happen.
When the church is complete, the Lord Jesus will take her to himself and purify her; she will be his bride. He will then assume his office as king. How?
He will return to the earth. His first advent was mild, but it shook Jerusalem. His second advent will be the most dramatic event in human history. He will force his way through, and set up his kingdom. Most humans will wail in terror when they see him, but the righteous Jews will acknowledge him as their long rejected Messiah. He will rescue them from the rest of the world who will be hellbent on their "final solution" of destroying Israel - and be the Sun of righteousness who puts an end to the night of antisemitism and brings in the bright day of healing for Israel (Malachi 4:2).
The Lord Jesus' reign will run for a thousand years from Jerusalem, and then continue onto eternity in the new universe that God will create.