Note: I will be talking primarily about the anime community in this post and most of the terms used in this post will consequently reflect that, but I'm assuming it refers to reviewers of other media as well. If you want to replace every usage of "show" or "series" with "game" or "franchise" in your mind, please do so.
Part 1: On Why People Are Doing Reviews Wrong
The popular review style in the anime community seems to be to give a number for story, art, sound, characters, and enjoyment, and then average these scores into an overall score. I have no clue where exactly people came up with this format. I suspect that they stole it from Tristan "Arkada" Gallant, whose "Glass Reflections" is one of the more popular shows out there. If I recall correctly, Arkada himself swiped that from a friend, but said friend may have gotten it somewhere else. So I don't really know who came up with it. It doesn't really matter, because the entire system is bogus and bullshit anyway.
Reviews are always going to be tricky things, because they attempt to give an objective value to things that are generally fairly subjective. Maybe you hate a show's art or animation style, so you dock it points for that. But there are other people who might love it. Inferno Cop has atrocious animation, but for some, that's the appeal of the show. I think that the character designs in Seitokai Yakuindomo look terrible, but I'm sure that some people like them. Humor is a similarly subjective thing. To go back to Seitokai Yakuindomo, I've never gotten past an episode or two because I feel like just having "haha, sex!" as the punchline isn't clever and is just a cheap way of getting laughs. Other people love the show, though, so my opinion isn't going to matter to them. Every one of those categories is going to involve subjectivity. Not everyone is going to agree on everything.
And yet, those categories are often scored on 10-point--or even 100-point--scales (note: they will try to hide the fact that they are 100-point scales by using decimals. Do not be fooled. If the score includes an 8.1, that's actually 81/100). They use these scales because, presumably, the more granular your score is, the more accurate it must be. If you're giving a show an 8.3 instead of an 8.2, people assume you've done some serious research. After all, that extra .1 point must have come from somewhere. There's no way that the decision was arbitrary, right?
This is the biggest lie that review scores tell. The thing that both reviewers and people who read reviews seem to be unaware of is that a larger review scale makes scores less objective, not more. By adding more nuance to scoring, you introduce more subjectivity.
I've always hated 10-point scales because they feel arbitrary to me. In practice, people generally only use a few of the numbers anyway. Instead of 5/10 being average, anything that gets a 5/10 is generally seen as really bad. Everyone has a different definition of what a 10-point scale means. Sometimes, people will be willing to use all 10 numbers, at which point they get called out for being too strict. Other times, they inflate scores and avoid giving anything below a 7, which is how you end up with all those "10/10, it's okay" jokes that people add at the end of their reviews on Steam or what have you.
So stop doing this thing where you divide shows into several elements and give them all subjective ratings. All you're doing is making the review less clear and less cohesive. If you tell me that a show's story gets an 8, its art gets a 7, its sound gets a 6, the characters get a 7, and your personal enjoyment is an 8, that tells me literally nothing about the show. If the soundtrack is lackluster, does that mean that the rest of the elements are dragged down by it? Are entertaining characters enough to save a boring plot? None of these scores tell me whether the show as a whole is good or bad, so averaging them for some grand final score isn't going to tell me anything either.
Part 2: On How To Evaluate A Series Correctly
I think that a good reviewer will approach every series a bit differently, because not all shows can be compared to each other. Some of my favorite anime include JoJo's Bizarre Adventure, The Tatami Galaxy, Silver Spoon, and Nichijou, but I love all of them for wildly different reasons and I can't really say "this one is better than this one because it did X better."
Ultimately, to determine a show's quality, you need to start with this key question: What is this show trying to do, and did the show do it correctly? First, identify the purpose of a show. Is it supposed to make you think? Does it exist to provide an emotional, cathartic release? Is it just meant to be fun and entertaining? Once you've identified the purpose, evaluate how well it's able to live up to that. This is probably going to be a good indicator of what your final verdict is going to be.
After that, don't look at the same elements every time and give them a grade. Look at the show's strengths and look at its weaknesses. What's going to draw people to the show, and what's going to turn them away? And since these aren't always mutually exclusive, be sure to look at the elements that could be hit-or-miss. The fourth season of Arrested Development, for example, drastically changed the structure so that each episode focused on a single character, showing what they had been up to for the past few years and only gradually revealing how they were related. Some people (myself included) thought that this was a fantastic and innovative structure. Others hated it.
Reviews can be structured, but I feel like this hampers a review. It backs you into a corner that you won't allow yourself to deviate from. If you're only going to focus on certain types of shows, that's one thing, and with something like a video game, I understand that gameplay, story, visuals, etc. should all probably at least be touched on. But if you limit your structure, you're limiting your ability to experiment. I enjoy different things for different reasons. Tengen Toppa Gurren Lagann is exciting and inspiring. Mushishi is ethereal and calming. You need to bring those attitudes with you into your review.
In addition, you'll also have biases, and you need to bring those into your review as well. Yes, it means you're not being as objective, but you're going to have biases anyway, so you may as well be up front about them. I can't explain why I think that Silver Spoon is so great without touching on how I grew up in a farming community or that I spent time in rural Hokkaido. I can't talk about Welcome to the NHK without explaining how I was in a situation uncomfortably close to the protagonist's and how the message of "if you continue down this downward path, necessity will eventually force you to get a job or die" that the show gave me was actually somehow disturbingly encouraging and exactly what I needed to hear.
Every show is different, so you need to approach every show differently. Maybe a popular show needs to be taken down a peg. Maybe you need to hype up an underappreciated gem. Whatever it is, don't limit yourself to a restrictive style. Especially when that style is stolen from other people and requires you to give scores you don't know how to give to aspects that you don't know much about. Seriously, you don't want me talking about animation and direction because my reasoning for a score will always boil down to "I dunno, I thought it looked good."
Part 3: On Alternative Methods of Scoring
No, seriously, you don't have to give something a score. All you need to do is explain what does or doesn't work. How you talk about a show is more important than actually scoring it. This also means that you have to actually have substance to your reviews, though. I've seen certain reviewers talk for minutes without actually saying anything of value.
But hey, some people feel like they can't break free of a rating system, so here are some other options.
I've seen a lot of scales represented in more practical terms,
such as "buy it, rent/stream it, skip it." I like ideas like that. But
no matter what you end up doing, make sure that you, at very least, have
some sort of rubric that you judge shows on. Have a way to describe each of your ratings and why a show would fit into them. Know who you would recommend a show to and why. Know who you wouldn't recommend it to.
As for systems themselves, start with a three-point system. Define things as good, neutral, or bad. If that's not something that works for you, expand it to a five-point system (the one I personally use) so that you can reward something that has that masterpiece quality to it, or to condemn something for being amateurish or downright offensive.
If you really want to add more nuance, try letter grades. That preserves the 5-point "terrible, bad, average, good, fantastic" scale in the form of A, B, C, D, and F, but it also allows you to add a + or - to say "I'm a bit on the fence about my rating."
If you're forced into a system you don't like, tweak a system that you do like so that it applies to it. Sometimes I do reviews on sites that force me to give a rating on a 100-point scale. I personally scale it so that I'm rating by intervals of 5, removing 100 (I could give a perfect score if I wanted but on a scale that granular I don't feel comfortable doing so) and 0 (everything at least deserves 5 points for existing). Then I divide it so that I know roughly where each series is going to fall. For example, 70-75 defines a good show that more people than not will enjoy, while 40-45 is more of a "guilty pleasure" show. I use benchmarks like this to define my entire scale, making it easier to give a rating that isn't as arbitrary.
In review (pun intended), don't make your scale bigger because you think it will make the review more objective. Don't feel like you have to structure yourself. Write a review because it's the right approach to take with a show. Look at the show relative to its peers rather than some nonexistant universal constant.
And please, whatever you do, don't score shows by averaging six categories. Seriously. That's stupid. Stop it.