I do not personally work for McKinsey, but have a few friends who do and I also work in the consulting sector. First of all, it helps (but is of course NOT a prerequisite) if you study or graduated from a top university and have good grades. The major doesn’t matter, so you are not supposed to have a MBA. I think that the recruiting process is almost the same or at least similar in all big consulting companies. Apart from slight differences, the bigger companies in general are very similar in their work and their culture is usually the differentiating factor. Therefore, it’s crucial to research about McKinsey’s unique culture and how its values align well with your personality. Think of experiences that will convince them of your personality and its alignment with the firm's culture. In an interview you will definitely be asked “Why McKinsey?”.
Then, it’s very helpful to attend networking events, because consulting firms want to develop relationships as early as possible.
And of course, you should have a strong application, i.e. a strong cover letter and CV.
Here are two links to reports of people who got an offer at McKinsey:
When you finally got an interview at McKinsey you should definitely focus on the case interview part, because that’s where most candidates fail. If you’ve never heard of case interviews, here’s an explanation (click on case study):
There are plenty (online) resources that’ll help you prepare for the interview (including the link above).