You Should Begin Your Resume with Your Educational Background
First, unless you have been out of school for ten (or more) years, it is always best to begin your resume with your education. In addition, you should always tout any honors or accolades that you received either at law school or your undergraduate institution. If you graduated summa cum laude or in the top 10%, then by all means include that information on your resume in a way that the reader will notice. The one exception to this rule is if you went to a second or third tier law school, but you are currently working with a top firm. In this situation, you may want to begin your resume with your "Experience" section, as the name of your firm is likely to pique the interest of the reader.
Stress Your Skill Set that Matches the New Employer's Needs
Second, it is very important to figure out the potential employer's needs and tailor your resume accordingly. For example, if you know that the potential new employer has a very active capital markets practice, then you should stress your experience in that area. The foregoing may sound obvious, but I often work with candidates who have broad experience and they fail to tailor their resume for different opportunities. If you have experience that is a very good match, then you may want to consider emphasizing that experience in your resume, even if it means omitting some of your other very good but perhaps irrelevant experience.
Streamline Your Resume
Third, it is imperative that you resume NOT be cluttered with a lot of information. Unless you are a very recent graduate from law school, it is probably best to stick with your education, current and past employment rather than keep all your internships, etc. On occasion I tell a candidate to keep an internship if it is very unusual or it has sparked great conversation in past interviews. However, the general rule is to streamline your resume, and to keep it to one page. A candidate who is more senior and has significant deal experience and/or has published extensively can reference a deal sheet or a summary of publications on his resume; however, the actual resume should be kept to one page whenever possible.
Proofread Your Resume Carefully
Fourth (and I feel a little silly even mentioning this matter) you need to proof your resume carefully and make sure that there are no typographical errors or additional spaces. You need to understand that the reader is generally looking for a way to winnow the pool to the absolute best group of candidates; and a candidate who cannot present a perfect one page document - the resume - is probably not going to be included in said group.
Use Your Cover Letter to Impart Personal Details
I have focused the bulk of this article on presenting the best resume possible, but I do have some advice re: cover letters. A cover letter is your opportunity to present yourself as a person. The more energetic and authentic you can be in your cover letter, the more you increase the chances that a potential new employer will want to meet you. Depending upon the situation, you may want to close your letter with a statement that you will get in touch with them "next week". I realize that the foregoing may seem a bit aggressive, but the worst thing that can happen is that when you call, you merely leave a message. The very best that can happen is that you actually get to speak with your potential new employer and demonstrate that you are articulate and motivated.
In order to separate yourself from the other candidates, you must present a strong resume that clearly demonstrates why you are the right candidate for the position. You can use your cover letter to convey personal details, and do not be afraid to be aggressive. The market is difficult and you should make every effort to show your potential new employer that you are very interested in working with them.