Here are some answers:
Rashi suggests that Haman had considered himself a god, and therefore, Mordechai refused to bow to him as per the commandment not to bow to "other gods". (See Josephus and Esther Rabbah 7:5)
Ibn Ezra suggests that Haman had idols hanging off his clothing, so that the people bowing to him would be essentially bowing to idols. (See Talmud Bavli Megilla 19)
According to these views, the reason he did not bow was because he was Jewish.
Alternatively, the fact that he had told them that he was a Jew was unrelated to why he didn't bow, and he didn't bow for other reasons. Bowing is traditionally a sign of respect, so one easy interpretation would be that Mordechai may have felt that Haman did not deserve his respect. This is not contingent on any other presumptions or extra-biblical material.
According to this understanding, the fact that Mordechai was a Jew was unrelated to his refusal to bow, something which is also a plausible reading of the narration here.