As a matter of fact, it appears from two etymological entries (in Etymonline.com) that "James" comes not from "Joshua" but from "Jacob":
masc. proper name, name of two of Christ's disciples, late 12c. Middle English vernacular form of Late Latin Jacomus (source of Old French James, Spanish Jaime, Italian Giacomo), altered from Latin Jacobus (see Jacob).
The entry on Jacob adds:
masc. proper name, name of Old Testament patriarch, son of Isaac and Rebecca and father of the founders of the twelve tribes, from Late Latin Iacobus, from Greek Iakobos, from Hebrew Ya'aqobh, literally "one that takes by the heel" (Gen. xxviii:12), a derivative of 'aqebh "heel."
Looking at the beginning of the letter of James, I see:
Ἰάκωβος Θεοῦ καὶ Κυρίου Ἰησοῦ Χριστοῦ δοῦλος ταῖς δώδεκα φυλαῖς ταῖς ἐν τῇ Διασπορᾷ χαίρειν.
that is (transliterated):
Iakobos Theou kai Kyriou Iesou Christou doulos tais dodeka phylais tais en te Diaspora khairein
Iakobos a servant of God and of the Lord Jesus Christ to the twelve tribes scattered among the nations: Greetings.
It therefore seems that King James did not put his name into the Bible; he simply happened to have (accounting for transformations) the same name as the (nominal) author of the letter.