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.
There are numerous reasons given for Bigsan and Seresh’s anger. The Yalkut Shimoni (1053) says that Bigsan and Seresh previously had important positions, and were upset with Mordechai for seemingly usurping them. The Malbim sees this in the actual words of the verse. After all, the verse relates Mordechai’s “sitting at the king’s gate” to Bigsan and Seresh’s anger to point out that their anger was directly caused by his position.
The Talmud (Megillah 13b) says they were upset with the king and queen because they were tired at nights having to protect their door as they spent time together, much like Pharoah became upset with the baker and butler for small reasons (Bireishis 40:1).
Rav Elisha Gallico says they were upset that Mordechai was sitting at the king’s gate, deciding cases based on Jewish law. Bigsan and Seresh therefore saw it as a patriotic duty to kill the king (who was, by the way, not Persian) for his seeming betrayal of Persian law in promoting a Jew to this position.
The Me’am Loez writes that the two of them were relatives of Vashti, and waited this long period of time to avenge her death. Another opinion he brings is that they were upset that Achashverosh rejected Haman’s daughter during the search for a new queen, and Haman convinced them to join in a rebellion against the king. The Me’am Loez also quotes Yossipon that this was just one part of a much larger rebellion, and that these two wanted to kill Achashverosh to bring the king’s head to the Greeks, enemies of the Persians, and thereby ingratiate themselves to them.
The Malbim points out that, regardless of the reason, their motivation was petty. This parallels the story of Yosef mentioned in the above Talmud. The Malbim continues that, just as H-Shem can inspire a king (Pharoah) to be angry with his servants for no reason, so, too, can H-Shem inspire a servant (Bigsan and Seresh) to hate a king for no reason. Rav Dovid Feinstein asks why these two stories are being paralleled by the Talmud. He answers that the verse says “shnei” (“both”), equating the feelings of Bigsan and Seresh. In the natural order of things, this is impossible because no two people can have the same exact emotion from the same exact motivation in the same exact degree. This is yet one more indication that these events were led Divinely.