If it pleases the King, let there go a forth a royal edict from him... that every man should rule in his own home." (1:19-22)
The King asked Haman only what to do with Vashti; why did Haman offer additional advice?
It was the custom in Persia that the King not decide any issue on his own; instead he would gather his advisers and seek their opinion (see Esther 1:13). Haman decided that this should be changed. Therefore, in addition to telling the King to kill Vashti, he suggested that from then on Achashveirosh should make all decisions on his own, without leaving room for appeal. Additionally, from then on every man should be the ruler of his home.
Everything recorded in the Megillah is connected to the Miracle of Purim. Were it not for these two recommendations made by Haman, Esther would not have become queen, and Haman would not have been hung.
After Vashti was killed, a search began for a new Queen. Had it not been the rule of the land that each man be the ruler of his home, when agents would have arrived at the home of Mordechai searching for Esther, he would have told them, "I don't know where Esther is. She left without my permission and did not say when she would be back." Thanks to Haman's advice, Mordechai was unable to hide Esther and thus she was forced to come to Achashveirosh and eventually become the queen.
When Achashveirosh heard from Charvonah that Haman had prepared gallows upon which to hang Mordechai, the King angrily said, "Hang him on it!" (7:9) Haman began to demand, "Before you hang me there must be a trial with a jury deciding if I am guilty. " Achashveirosh told Haman, "Sorry! It was you who gave the advice some time ago that 'yeitzei devar malchut milfanav' -- the King should make decisions on his own and no one can appeal them." Thus, Haman was hung immediately.