Admittedly, there are some countries where PhD students are in effect hired by a faculty member and selected by them, but that's not how it's done here. At Stanford, a committee examines all applications and picks the ones with the most promise for study in Computer Science. The process is honest and objective. It is impossible to influence it through individual faculty, regardless of whether or not they are on the committee. In fact, I know of some faculty who will count it against you if you send this sort of email.
Whenever I get a "please treat me specially" letter, I respond with the following:
Thank you for your interest in Stanford. All admissions decisions are made by a committee of faculty and graduate students, and there is no point contacting individual faculty in the hope of bringing your case to their attention. Questions regarding admissions should be sent to admissions @ cs.stanford.edu You may also find out more about our department from URL http://www-cs.stanford.edu
Added Nov., 2003: A more recent variation is people asking for "summer internships." Unfortunately, the same principle applies. You can't get a student job at a university without being a student at that university. Faculty have a responsibility to serve the needs of the students at the school that pays their salary.
The second sort of email I get a lot of sounds like: "I really like your book on [fill in the blank], and I'm learning a lot. But I just have a doubt about the solution to Exercise 4.5.6. Could you please tell me the answer?" In a slight variant, it's not a book exercise, but another problem whose solution they would like to see. I'm not that stupid. The chances are 98% that this is a homework assignment, and I'm not going to do your work for you. I therefore have developed a policy of responding to questions about material in my books only to bona-fide instructors. I rarely hear back when these email writers get the following stock response:
Thank you for your note. When I get these sorts of questions, I like to know first what school you are attending, what class you are taking, and who the instructor is (email if possible please). I suggest that first you discuss the problem with your local instructor. If they can't help you, then please ask them to get in touch with me.