Contact your country's binational Fulbright commission to obtain an application and information on eligibility criteria. Or, if your country doesn't have a binational commission, contact the U.S. embassy in your country.
Make sure you fit the criteria. Though they differ from country to country, you must at least be a citizen of a country other than the United States at the time of application.
Think precisely about what sort of program you'd like to design for yourself and about how you'd describe the project to the funding committee.
Consider designing your program around the theme of mutual understanding between nations, as this is what the fellowship exists to promote.
Notify professors, former employers, etc., well ahead of time for any letters you need from them.
Send in all the materials before the deadline. Again, dates vary by country, but apply well before the application deadline to avoid the risk of missing it.