"Tell me about yourself" is the official start of the interview. The best way to answer it is to tell the interviewer general things about you that will matter to them as they consider you for the position.
This is a perfect opportunity for you to set yourself above all other candidates. Its open-ended nature allows you the freedom to say anything you want. So, think about the job, what it entails, and all the research you have done. Put yourself in the hiring manager's shoes: what about your experiences, skills, or interests are the most impressive? What can you say to get that person's attention so they will sit up and take notice of you?
Here's how to structure the best answer for "Tell me about yourself": education, background, key highlights. All in a brief, minute-or-two answer.
Perhaps start with your education—what's your degree? If you had a particularly high GPA, you might mention it. Did you take specific classes that pertain to this job? Talk about them.
After education, you want to talk a bit about your background and employment history. Hit the highlights: promotions, awards, or key accomplishments. These might not necessarily be the things you're most proud of, but they should be the things that would most impress this hiring manager.
All this requires some strategic thinking on your part before your interview, but liken it to creating and tailoring your resume. Your resume is written specifically for each job before it is sent; this question should be answered specifically for this interview. Just ask yourself this question: What part of my story will impress this hiring manager so I will be moved up on the list of consideration for hire? Get used to strategic thinking and plan for it, because that's what a lot of interview questions are about.
Don't take more than a minute or so to answer the question: just deliver a targeted message that tells the hiring manager: "I am skilled; I have some impressive accomplishments and I want to bring my talents to work here for you."