Figure Analysis
What exactly is this dude thinking? It's a tough question, we admit. King Ahasuerus agrees to help commit a genocide simply because his wicked counselor suggests doing so. He doesn't seem to give it too much thought, but later, when he offers Esther a boon, she's able to reverse the whole process. She tells Ahasuerus that Haman is wicked and that he (Ahasuerus) should save the Jews. So Ahasuerus rises up in wrath and has Haman executed. But Esther doesn't, apparently, tell Ahasuerus anything he didn't already know (other than revealing that she herself is a Jew, which is obviously an important piece of information). She just informs him that Haman is wicked and needs to go.
It doesn't seem like Ahasuerus is coming to some big realization or even becoming a better person, one who is capable of seeing that genocide is wrong and that the Jews worship a real God. He just kind of casually agrees with whoever happens to be in the position to advise him. He doesn't necessarily grow as a character. To the contrary, that's the kind of process that kings like Nebuchadnezzar and Darius the Mede go through in Daniel—not so, in this case.
Thus, it's hard to say if Ahasuerus has progressed as a person or improved morally. You could definitely say that about Nebuchadnezzar in Daniel. But in this book it seems to be mainly the gift he promises to Esther that reverses his position on murdering all the Jews. It wasn't a decision based on his own feelings in the first place, and it's based more on what Esther's telling him to do now.
Weak Sauce
Taking all this into account, Ahasuerus seems kind of weak-minded. Unlike his noble forebear, King Cyrus the Great (who is identified by the Bible as a kind of messiah), Ahasuerus seems to be at the whim of the people around him. He's a large order of lame duck, covered in weak sauce.
When he's under Haman's influence, he does what Haman wants him to do. When he's under Esther's influence, he does what Esther wants him to do. Also, he performs occasionally disrespectful and boorish antics like trying to "show off" his former Queen Vashti to his people, presumably in a lewd manner (which is why she refuses to show up—again, presumably).
Progressively Weaker Sauce
The Canadian literary critic Northrop Frye said the Gentile kings in the Bible usually start off being relatively good (or, at least, not bad), while their descendants degenerate into being worse: weak, ineffectual, violent. The pharaoh who rewards Joseph in Genesis gives way to the pharaoh who tries to massacre the Hebrew infants in Exodus.
In Daniel, the glorious (though initially confused and rather insane) King Nebuchadnezzar gives way to the useless milk-sop, Belshazzar. The same pattern happens with Greek Kings like (according to Jewish tradition) the benevolent Alexander the Great, who (after a few more kings) is succeeded by a vicious, genocidal maniac, Antioch Epiphanes VI. The same pattern plays out here, with the truly boss Cyrus the Great and the good king Darius being succeeded by a pretty lousy and conscience-less king like Ahasuerus.
Also, Ahasuerus is supposedly based on (or perhaps the same as) the real, historical Persian King Xerxes I who makes a decidedly different kind of appearance in the hit movie, 300.