According to the Midrash (Esther Rabba 9:2), Zeresh was advising Haman to convince the king that it was imperative that he kill Mordechai. He was supposed to explain to Achashverosh that Mordechai’s very existence threatened Persian civilization, even to the point that just his praying had the power to negate the power of their Persian gods.
For another explanation for Haman’s not needing to ask the king, but rather tell him, we should recall that the Yalkut Shimoni (1053, commenting on Esther 3:1) writes that Achashverosh had built a throne for Haman that was even higher than his own, and that the Talmud (Megillah 51a, commenting on Esther 4:1) says that Mordechai’s cry after learning of the decree against the Jews was that Haman became more powerful than Achashverosh. Based on these, the Avnei Nezer writes that Haman was to tell the king instead of asking him because he had taken charge. Perhaps, as in history or literature, the king gave his adviser greater power than he realized in order to have less responsibility, himself.