It depends on what you mean by "becoming a medievalist"!

If you mean a professional whose career revolved around studying or communicating the Middle Ages--as a professor, curator, manuscript assessor for an auction house, research librarians, etc (and note that these positions are extremely competitive)--then the only real way to get there is a PhD in history or an allied field with a specialty in a medieval subfield. "Allied field" most likely means a national literature (e.g. English, German, Near Eastern studies), but it could also be theology, archaeology, art history, etc.

If your goal is a PhD in a medieval field, you should start aiming yourself there in undergrad. You can do it otherwise--I did--but it's harder and I don't recommend it. The most important thing to start lining up is languages. If you want to work on western Europe, you WILL need Latin, and you will need STRONG Latin. Preferably 3-4 years in undergrad. If you're past that point, there are some schools that offer an intensive "post-bacc" year in classics where you study basically Latin and Greek for a year. It's not cheap, but I really wish I had done this. Medievalists are also conversant in vernacular medieval languages and modern scholarly languages. It's standard for a PhD program to require French and German, but allow the substitution of a different regional language if that's what you study (e.g. you study late medieval Spain? Do Spanish instead of German). Especially if you're not planning to study later medieval England, you want a solid foundation in your regional language by high school or undergrad for sure. Also, if you plan to study Christian-Muslim or Christian-Jewish interactions, or the Islamic world, Arabic or Hebrew must be in the docket. For Arabic, you may be able to get government funding for summer study as a critical-need language, especially for programs abroad.

At the same time, you'll want to make sure your undergrad coursework involves medieval-focused classes. Because a PhD--and entrance to a PhD program--requires fairly intensive specialization within "medieval" and even within something like "medieval English Church history," you want to gain a broad sense of the topics that are out there, and what interests you enough to spend 2-5 years of your life zeroed in on a single subject. Advanced undergraduate coursework in history or your chosen subject, meanwhile, will start to familiarize you with what professional historians (or &c) do as far as theoretical underpinnings and methodology go. You might also try to get some experience in paleography, which is to say, studying and flat-out reading medieval manuscripts/handwriting.

You should try to position yourself to write a senior thesis as well, honors or not. It will be a good test of whether a long-term research project and major, major paper is something you really, truly enjoy enough to spend your life doing.

Due to the intense language, paleography, and knowledge demands of medieval, a lot of incoming PhD students in this subfield wind up with a master's degree first. Try to find one that is funded, either through a stipend or where they allow you to TA for pay.

If, on the other hand, by "become a medievalist" you mean someone who loves the Middle Ages and knows a ton about them (say...earns medieval flair in AskHistorians? :D ), the best thing to do is READ. READ READ READ. We have an excellent book list to get you started on some recommendations, to take you down a path you're interested in.

A really great way to dig into medieval (or any subfield!) is to Google around for "phd exam list history [field]." PhD comprehensive/qualifying/general exams usually involve reading 25-100 books and articles focused on a single topic, like "medieval and early modern Mediterranean history." Don't get bogged down in the bare facts while you read. Reading that many books, you will absorb the facts eventually, trust me. Rather, pay attention to the overall statements that the authors are making about history, to their arguments. (Especially pay attention to the books' introductions and, if they have them, conclusion).

Basically: learn Latin, get really really interested in a narrow topic, and read your heart out!