It’s résumé I présumé
It turns out the punctuation of the term résumé is by no means settled law. All three popular options: resume, resumé and résumé are in the dictionary, so it doesn’t really matter much. Coming mostly from a PR/journalism background, however, I defer to the Associated Press styles guide on matters such as these and go for résumé. If you read the Wall Street Journal, New York Times, or most newspapers in America, you will find it with all of the original French accent marks.
Also, on this Web site (AP punctuation of Web site btw) , I couldn’t very well ignore accent marks with the sub-title “setting your career rhythm” and a guy waving a baton on the first page.
As for whether you use the French accents on your own résumé, don’t even use the word at all. Just put your name at the top of the document.
For the cover letter you have three options, write around it, use the accents, which I would recommend, or if you can find out what that particular employer does, just do that.