You have nothing to worry about regarding your accent. There are plenty of thick accents in law school. You might even have an advantage in getting into law school because the schools want people from different backgrounds. Being from another country is definitly a plus. (I'm assuming your from another country. If you're not, you can still make note that english is not your first language and that should help you because it shows you've overcome a language obsticle.)
A couple things:
You might need to work extra hard on your personal statment and your writing skills in general. Lawyers don't tolerate grammer and spelling mistakes (haha - I'm sure this post is full of them, but no clients, bosses, or judges are going to be reading this).
You should keep in mind that you will get called on in law school, and professors and other students will expect you to do a good job answering the professor's questions. Some professors go easy on people with accents, while others will treat you no different than your classmates. How you speak is not important, what matters is what you say. Study hard and don't be intimidated when its your turn to talk. You'll be just fine.
I don't know the implications of citizenship on law school admissions, but I know it's easier to get financial aid (because the US gov't provides student loans, and most private loans require either US citizenship or a US cosigner).