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.
The Midrash (Esther Rabba 9:2) writes that Zeresh and Haman’s other advisers recommended that Haman hang Mordechai because this is a form of death from which H-Shem had never previously saved the Jews miraculously.
Seeing that they were, however, saved from every other type of death, it seems strange that Zeresh could so grossly misunderstand history. Aruchas Tamid explains that Zeresh thought that the Jews had previously been rescued by using magic. Therefore, Mordechai would be unable to escape hanging since the Talmud (Sanhedrin 44b, Rashi) teaches that magicians need to be standing on the ground to perform their magic. In fact, this is one of the reasons Pharoah’s magicians could not replicate the plague of lice (Shemos 8:14, Da’as Zekeinim Baalei Tosfos), since the lice covered the ground and the magicians could not stand on it. For this reason, in the famous aggadic story in the Talmud when Shimon ben Shetach killed eight magicians (Sanhedrin 45b), he first lifted them from the ground.
M’nos HaLevi says that they wanted Mordechai hanged in order to avenge the hanging deaths of Bigsan and Seresh (Esther 2:23), Haman’s friends and possibly co-conspirators. R’ Yehonasan Eibshutz reiterates that killing Mordechai was Haman’s first step in killing Achashverosh and Esther, and becoming king through his friendship with the Greeks, rivals of Persia.