"Then the royal officials at the king's gate asked Mordecai, 'Why do you disobey the king's command?' Day after day they spoke to him but he refused to comply. Therefore they told Haman about it to see whether Mordecai's behavior would be tolerated, for he had told them he was a Jew." NIV translation
This is a good example of what we would call peer pressure. The officials tried to get Mordecai to go along and do what he knew was wrong but he held fast to the Word of God. Therefore, his "friends" turned him in to see what would happen. Mordecai had not had the faith to obey God and return to Jerusalem but he does have the faith to stand firm on this point.
"When Haman saw that Mordecai would not kneel down or pay him honor, he was enraged. Yet having learned who Mordecai's people were, he scorned the idea of killing only Mordecai. Instead Haman looked for a way to destroy all Mordecai's people, the Jews, throughout the whole kingdom of Xerxes." NIV translation
The idea that someone would not bow down to him was bad but to have a Jew refuse to honor him sent Haman over the top. He had been raised with this anti-semitism (hatred of the Jewish people) and it was just waiting to explode. The same type of thing is happening in many places around the world today as children are raised to hate God's people (the Jews).
We can take comfort in the fact that they (and we as Christians) are God's people and He takes care of His own. (see Revelation 11)
"In the twelfth year of King Xerxes, in the first month, the month of Nisan, they cast the pur (that is, the lot) in the presence of Haman to select a day and month. And the lot fell on the twelfth month, the month of Adar." NIV translation
So, they needed to decide on a date for the mass killing of the Jews. To pick a date, they cast lots and the date was revealed to them.
"Then Haman said to King Xerxes, 'There is a certain people dispersed and scattered among the peoples in all the provinces of your kingdom whose customs are different from those of all other people and who do not obey the king's laws; it is not in the king's best interest to tolerate them. If it pleases the king, let a decree be issued to destroy them, and I will put ten thousand talents of silver into the royal treasury for the men who carry out this business.'" NIV translation
Haman knows that he cannot eliminate them on his own, so he looks to the king for help. He appeals to the king's fear of losing power to get a law to authorize the killing. Haman even offers a reward for the men that will do the killing. The same sort of tactics are being used today to attack Israel in that those that actually hate the most pay others to do the killing. Some "religious leader" will issue a decree and offer money so that some poor young man will do his dirty work.