Get an undergraduate and master's degree in a related field, such as biology. Not all schools will require a master's degree, and not all schools will require the same undergraduate courses or majors. You'll have to check individual school requirements to see if your credentials are sufficient.
Choose one or more schools that you are interested in applying to. Most research can be done online, but you can also call or visit schools. Some magazines print special editions in which schools and programs are ranked. Most schools offer tours or the chance to meet one-on-one with admissions officers, professors and even students. Admissions deadlines, requirements and procedures will vary from institution to institution and from country to country --- sometimes widely. So it's important that when choosing your schools you pick only those that you are already qualified to apply to or will be qualified to apply to before the application deadline.
Arrange your application materials. Again, required materials will vary from school to school and country to country, but most will require prior degrees, GRE scores or equivalents (though these scores may be weighted differently from one institution to the next), a personal statement and letters of recommendation.
Apply. This can be done either online or, sometimes, by mail, though many institutions only accept online applications. Each institution will have its own procedure and set of deadlines, so pay close attention to the specific instructions for each school to which you are applying. Most schools start accepting applications in the fall and close the process in January or February.
Accept your offer of admission. If your application is accepted, you'll usually have to formally accept your admission. This can often be done via email, though some schools may require an official form. If you are not accepted, the school may or may not state why. In this case, you may either apply to other institutions or work to improve your qualifications (for example, through higher grades or better entrance-test scores).
Study. A lot. Once you arrive on campus you'll need to fulfill a number of requirements such as courses, lab work, exams, and a dissertation. This is where the real work comes in, and it will likely take four or more years to finish your doctorate. Again, requirements will vary from school to school, but they will generally resemble one another.