How do You become a video game tester?
I'm a patent attorney for a video game company, which means I have to play video games in order to understand what is new and potentially protectable. It's a fun job, took me a while to get here – I got a degree in chemistry, then worked as a research chemist, then a pharmaceutical consultant, then went to law school, then worked at a law firm, then worked at a tech company, before being fortunate enough to land my current job. But you're probably thinking about a different type of career path.
My brother was a researcher at a life sciences company with a salary and benefits and job security, but he quit to take a minimum wage jobs answering phones at a large video game company, and eventually worked his way up to testing video games, worked in tech, then recently was head of engineering at a small gaming company.
Either way, there is probably no one true path, but the real question is what you want to do with your gaming skills. Do you want to design games? Code games? Play games professionally? Or just look for bugs? If it's the latter (which is usually what folks mean when they want to test games), then consider that it's not all fun and games: http://www.businessinsider.com/what-its-like-to-be-a-video-game-tester-2015-6
On the other hand, if you play a lot, and play with friends, then you may be building up a fair store of knowledge on what makes a game good, retains players, and ultimately makes money. In which case, the video game industry has a lot of opportunities available. It's not an easy industry by any means, but it sure is fun!
If you're really good, and somewhat entertaining, you may consider streaming on Twitch. Or providing reviews on YouTube. If you're good enough at MMOs (e.g., DOTA, LoL, HoS, HS) to compete, then consider going into eSports, which some universities are actually recruiting for. If you just love video games and the industry, and want to be a part of it, while still playing games, then consider coupling your passion with another discipline, like computer science, game design, business, law, finance, or others.
And don't let anyone tell you that playing video games is a complete waste of time. It worked out for this guy: http://www.theverge.com/2015/5/22/8643065/mark-zuckerberg-video-games-good-for-kids
But also remember that balance is important in life.
What I did was playtest and network. Check and see what game studios are close to you and sign up for playtesting with them and attend each and every session that you can. While there, do not act like you know everything, do not be creepy or overly complimentary. Keep an eye on the job postings that they have and apply each time they open a testing position. You never know when they may call.
The good news is that depending on where you live, it is actually very easy to become a video game tester. Some companies prefer that you have some background in game design or computer science, but many only require good communication skills, an attention to detail, and a minimum of a high school education. Often times just living in the area of the company you want to work for will be all the qualification you need.
The bad news is that game testing is not as nice as it might seem. Testers are often under-valued by their employers, and depending on the company can have poor relations with the design departments. They are very often underpaid and made to work long or inconsistent hours with no real guarantee of job security. The work itself can be tedious and not at all like just playing video games all day. Advancement in QA positions often lead to management type roles where you will spend less and less time actually "on the floor" testing games and more time managing other people.
Higher ranking QA positions can also be involved with working along side designers to write test passes (or "plans" for how to test one specific aspect of a game) that the lower ranking game testers will carry out. Other, better paying, more valued jobs that involve game testing which I would recommend looking in to if you are still interested in the field include:
Quality Assurance Engineers (or QAE) who are programmers that write applications which automate repetitive testing tasks. A degree in computer science or a strong programming background is a must to enter this profession.
Game-play Coordinators, which are more of a production role. They can supervise large scale testing events, such as coordinating a group of volunteer alpha testers to test wider systems that are outside of the scope of normal QA, and collect feedback as to how to improve the game-play experience. Coordinators may be able to work on a consultation basis (ie. they are not affiliated with any individual company, but jump from project to project throughout the industry) and as such can make a comfortable amount of money. A position like that is a lot harder to get in to though, and like many similar positions, require as much luck as experience. The job title is also a bit ambiguous, someone could have the same title and even the same responsibilities, but be in-house as opposed to a consultant, which makes their earnings much more modest.
There are likely many fringe cases that would afford you the ability to test games as part of your job, like Teddy Joe who commented here who is a patent attorney for games and entered the industry through law. Video games are a great amalgamation of many different forms of media, and as a result have many avenues of entry. The important thing that Teddy brought up that I just want to second is that testing games is not as glamorous as it sounds. If you get lucky with a company that has a good culture around game testing then it can be a fun and engaging job, but the majority of companies are not great places to work as a game tester. Really ask yourself, do you want to play games, or help make them? Because those are two very different things.