Should minority students approach the application process differently? How so?
By all means explain your family's history, but only your your race is unusual, such as Afro-Asian or Indo-Hispanic. A culture mix is appealing to admissions officers as it broadens the makeup of their student body.
Most minority students should not be afraid to embrace your ethnicity and your cultural background. Going to a majority white school can be intimidating if you have never learned in an environment in which students are different from you. If being from a particular cultural background provides a context for your achievement then talk about that. You should work to let the admission committee know who you are. If your cultural background and experience is an important aspect of your identity then you should talk about that in the process. I can show you exactly how to do that.