It is a very closely run issue, and as stated above can vary by school, but in general, the order goes:
- Letters of Recommendation
- Personal Statement
- GRE scores
If for some reason you don't get into the program of your choice you may consider applying to a U.S. Master's degree program that will get you up to speed, and make your application much more persuasive. Many schools offer 1 year masters in a variety of subjects, which are valuable in themselves, and are easier to get into for international students.
Letters of recommendation tell the committee a lot about you as a person, which is very important, since they are stuck with you for 5+ years. People coming from outside the U.S. are better off getting letters from scholars who have attended university in the U.S. because it is frequently difficult for U.S. scholars to weight the relative merits of courses, research and so forth from outside the country and someone intimately familiar with the U.S. system can help them parse that by making comparisons to things current in the U.S. Coming from the U.K. (or Australia, New Zealand, Canada, Germany and a few others) the systems are similar enough that the assistance is minor, but perhaps worth it all else being equal. Above all else make sure that your letter writers can speak in depth about you as a scholar and don't go for fame over substance.
The personal statement serves essentially the same purpose, but it shows the committee what you want to research while at the institution. The title "personal statement" is misleading, however, because you should treat it as a research statement. The biggest challenge students everywhere, but especially from overseas where different academic traditions may prevail, is completing a dissertation as a major research effort. Schools need to know that you have at least some idea what creating original scholarship means, and why you think going to their school is valuable to you as a scholar. They don't care about the time your dog died, and it inspired you to be a better person.
Courses show that you have the basic understanding of the topic, although unlike British or Continental programs, almost all U.S. PhD.'s include significant coursework, which will get you to where you need to be to write your dissertation. Grades don't necessarily compare across systems, and the committee knows that, but as long as you have done well and are prepared for the program.
GRE in almost every case is just a way to winnow out the thoroughly unqualified. FYI, most maths intensive disciplines (Math, Economics, many natural sciences) at top schools will not even consider you unless you have a perfect quant score (currently 170, formerly 800). Most schools don't hide their cutoffs, and if they do an e-mail to the department secretary will usually find it out for you, so don't waste your money applying to places you will never get in because you don't clear this obvious bar.
Finally, I highly recommend the book Succeeding as an International Student in the United States and Canada by Charles Lipson.