Here are some hints to consider, there are definitely more—but this is a start.
Be as specific and to the point as possible in terms of research direction and faculty you are interested in working with. Note how your experience or training can contribute to research currently being conducted or potential research that may be conducted once you arrive.
Submit a completed application as soon as possible, meeting the application deadline. Keep records, a file of every school you apply to.
Ask plenty of questions while preparing, during, and even after the application process. Know all your potential funding sources and how to access them.
Be assertive and do your research on schools and locations. Explore your options by gathering as much information about each school as possible and make a campus visit.
Expect opportunities to simply present themselves.
Submit anything without having made sure you have copies of the particular application component.