Before I respond, I would like to stress that a question of this sort -- an issue which, as you will soon see, is the subject of debate amongst halachic authorities -- must be addressed to your personal rabbi. That said, for information purposes I will give you a little background information on the subject.
As a general rule, dressing up in clothing exclusive to the opposite gender is definitely a problem. The Torah expressly forbids such behavior: "A man's attire shall not be on a woman, nor may a man wear a woman's garment."
This prohibition is intended to prevent licentious behavior. The question remains, however, whether this prohibition applies to Purim, a day when any cross-dressing would only be for "harmless" entertainment purposes. The Rama, in the Code of Jewish Law, mentions two opposite opinions on this question: 1. There is no prohibition. 2. It is prohibited. He then says that the custom is to follow the first, more lenient, opinion.
The Mishnah Berurah (halachic code authored by Rabbi Israel Meir Kagan (1838-1933)) cites several halachic authorities who recommend that the custom of Purim cross-dressing be abolished and prohibit this practice. Likewise, as far as I can see, the overwhelming preponderance of later rabbis who discuss this issue in their responsa is to prohibit the practice, citing other earlier authorities who call it "wicked."
But again, be sure to consult with your rabbi for an actual halachic decision.
Just two reminders: a) The costumes should be checked for shatnez before they are worn on Purim -- even if they are just rented. b) The laws of modesty apply on Purim as well... Be certain that any costume you wear conforms to the year-round modesty requirements.
Have a happy Purim!
Rabbi Eliezer Danzinger for Chabad.org