What are some things one can do outside of their job to accelerate their career in product marketing?

Such a great question. I think the best thing you can do invest in yourself and "sharpen the saw". This is especially critical earlier in your career. A couple of approaches: 

1.) Read - My mom has this saying: Books are like software updates for your brain. Yes, not all business books are spell-binding page turners, but you'll quickly see that your brain will free associate and you'll draw on differents nuggets, often at opportune times. I've stumbled on frameworks for example that helped chart the course for a GTM, or quotes that set the entire tone for a keynote. And it's not just business books, but stuff like sociology, psychology and even biographies. Fiction is also great because of the rich storytelling aspects, narrative arcs. If you find an author whose writing you love, read everything she's written because it will make you a better communicator. 

2.) Watch - Ted Talks, Superbowl ads, Saturday Night Live, it's all good--really. As a marketer I think it's really important to have your pulse on the zeitgeist and pop culture. This is advice I received from Danielle Morrill, an early Twilio employee, and while it felt self-indulgent and non-intuitive at first, it does help you come up with stuff that feels relatable. You're often just a good meme or pop culture reference away from making an idea stick. Podcasts are great for this too. 

3.) Speak - "Every time you speak, you are auditioning for leadership". This is a quote from James Humes, a former presidential speechwriter. The more you can dial into your authentic self and the earlier you get confident expressing your ideas in public settings, the better you will be at your job. People often have this reflexive "I suck at public speaking" fear, but all speaking is public speaking. Steve Jobs used to maniacally rehearse his keynotes. Marc Benioff starts his next big idea keynotes with a kernel that he first peppers into customer meetings or smaller settings and then continuously tweaks until it's Dreamforce ready. So do Toastmasters, or jump at opptys to represent your company at conferences. Speaking is a muscle and one that will make you stronger as a marketer. 

4. ) Courses: You can and should definitely take courses as well. Business school is obviously a popular option, and it helped moved the needle in my career as a career-switcher--but it's also expensive, which is why I started with some of the ideas above as they are more accessible and practical. Many employers offer e-learning options like Lynda, or in person classes as well. At Google, I goaled myself on taking at least one or two of these a quarter because to me, it felt like it was part of our compensation package--why wouldn't you do it?

Pragmattic marketing is excellent, I also recommend product marketing community events as well: http://productmarketingcommunity.com/

Also, practice makes perfect. If you have friends who need PMM help but can't afford to hire someone yet, offer to help them with their launches, messaging, etc. They will love it, you will get more experience, and it's a good resume builder to boot. Win / win! 

Become an excellent public speaker and networker. If you're not consistently getting 5/5's on your speaker evaluations, get some training or join a group like Toastmasters. I would probably master this skill before working on any others in my free time. 

2 things to add in addition to the responses already here: 

1. Be a visual communicator in addition to written and verbal. The old addage 'a picture is worth 1,000 words' applies so much to product marketing. It's not just what we say, it's how we say it, and what visuals are there to support and represent what you are trying to communicate. 
2. I'll add Sirius Decisions framework in addition to Pragmatic. I have done both, and depending on the company Sirius can be better. Good to know the interconnected nature of marketing, product management, and sales. 

I think Mike cov ered one of the most important skills, but there are certainly others. A few recommendations:

• Take a look at the Pragmatic Marketing courses. Most of their courses are geared towards product managers (PM), but they do have a "launch" course which is good. Also, even if you attend one of the more general PM-oriented classes it can give you valuable perspective in working with product teams.
• Look for events in your area! Depending on where you live there may be a ProductCamp, or a meetup near you that focuses on product marketing. Speaking from experience with ProductCamp it's a fantastic event where you can generally meet hundreds of product marketers in a short-period of time and likely get involved.
• Get certifications related to the industry or products you are already working on, or want to work on. For example, if you want to be a product marketer at a marketing software company it's a good idea to be familiar with software like Google Analytics. There are also free certifications from Google, HubSpot, and others that can help you get a foot in the door. 
• Read! There are countless books and blogs out in the market which are fantastic, but try to consume as much information as possible. After doing so for a little while you'll easily be able to filter out some really insightful advice from something that may not be worthwhile.

Great answers so far. I would also recommend chatting live with other PMM's to see how they advanced in their careers. There is so much context you get in a phone conversation or coffee meeting that you can't get via an online text discussion.

In case you want to see what types of people are available to meet, you can use Interviews.Tech 

Great tips so far. I want to piggy back off of Mike’s suggestion on practicing your public speaking through a group like Toastmasters. Toastmasters is a great organization with clubs in almost every city. While they generally have the same structure, the atmosphere may vary from club to club, so I’d suggest visiting a few to find the right fit. Not only will you get a chance to get comfortable with your public speaking, you’ll get feedback on how to improve with your storytelling, body language and engaging slide decks (which will be important for product marketing pitch decks).

Write. A lot. About different things. Become a master at using written communications (a blog post, an article, a LinkedIn post, an answer on here) to sell your point of view on something, whatever that may be. Because that's what you're doing in product marketing, whether you're writing for customers, analysts or internal stakeholders, you need to learn what's important to your audience to know, and how they best absorb it.