The Bible says nothing about any king named Joseph. The Jews had 23 kings, beginning with Saul and ending with Zedekiah. The ten tribes that broke away had 20 kings, beginning with Jeroboam and ending with Hoshea. Many foreign kings are mentioned by name. None of these kings is named Joseph.
But there is one person in the Bible who had the right to call himself King Joseph.
Anyone who has studied the Bible much is probably familiar with Isaiah 7:14:
Therefore the Lord himself will give you a sign. Behold, the virgin shall conceive and bear a son, and shall call his name Immanuel.
The New Testament tells us that it is a prophecy of the virgin birth of Jesus.