In answering this question, R’ Dovid Feinstein reminds us that the Talmud (Menachos 29b) says that H-Shem made Olam HaBa (the World to Come) using a yud. Therefore, Mordechai was saying that, in threatening the extermination of the Jews, the people who believed in the World to Come, Haman was actually attempting to subvert the authority of Heaven.
The Maharal writes that, by writing the word “baYihudim” with an extra letter “yud,” Mordechai was indicating the kind of Jews affected by Haman’s decree. The first yud represents the Jews in general, and the second yud represents those Jews willing to sacrifice their lives to sanctify G-d if the need arises.
The Tzemach Tzedek takes this idea one step further. In the beginning of Creation, H-Shem created within mankind two yetzarim (inclinations) (see Rashi to Bireishis 2:7). Haman, after all, did not discriminate – he wanted to kill all Jews, good or bad. Interestingly, later when the Jews are saved, the verse (Esther 9:15) spells out Yehudim with two yuds again to testify concerning them that the spirit of teshuva that enveloped the Jews at that time made it so the yetzer for evil had no affect on them.