Figure Analysis
King Ahaz is Hezekiah's dad. He seems to be a rather puzzled king in Isaiah. When he falls under threat of a joint-invasion by the northern Kingdom of Israel and by Ephraim, the Voice of God speaks to Ahaz, saying that it wants him to ask for a sign. Yet Ahaz thinks this is some sort of trial and refuses to ask, saying he's not going to put God to the test. Isaiah shows up to set the record straight. He says that God is going to send a sign, anyway: a child will be born to a virgin and grow up to eat honey and curds. Then Assyria will come and smash the joint-alliance, sacking Israel and Ephraim (see Isaiah: 7-9 for the primary source).
That's really the only role Ahaz plays in Isaiah. He helps facilitate this great prophecy. Additionally, his name "Ahaz" is an abbreviated version of "Jehoahaz" (like "Chris" instead of "Christopher," or maybe more like "Topher"), which means "The Lord Has Held."