Think about your positive argument first
Contrary to what the massively insecure and angry think, criticism isn't fundamentally, primarily or even largely a negative practice; it's analysis and contextualization. Inevitably, though, there'll come a time when something so wrongheaded pops up you feel the need to drop some verbal napalm. You don't need to include some kind of positive resolution at the end of your piece (indeed, doing so might — depending on what you're writing — just be the kind of hackneyed SAT essay "proper essay construction" fit only for, well, SAT essays). But you do need to be clear on why you're so fired-up; the answer, generally, is that whatever it is you're arguing against infuriates you because there's a better alternative: a better movie, a better argument for or against something, a better perspective, whatever. Once you know what you're implicitly or explicitly arguing for, it'll be easier to systematically demolish your targets.
Optional: Write fast and in the heat of the moment
There aren't any solid rules for How To Write — methods, times, sustained concentration — and anyone who tells you there are is just trying to mandate what works for them as perfect because it took them forever to figure out how best to work. However, I've found that when something just really ticks me off, it's best to sit down and get it all out of my system. Fast drafting retains your passion and allows you to barrel past all of the hemming and hawing that can clutter up a first draft. But this may not work for everyone.
Don't go ad hominem
This should probably go without saying, but apparently it doesn't. There are many reasons not to go ad hominem, such as a) it implies your argument isn't strong enough to rest on its own merits b) it leads you to make unflattering characterizations that may not be accurate c) it makes you a dick and life is hard enough d) you probably have mutual acquaintances or friends with the person you're attacking, which will make entering certain rooms tense e) you're potentially burning yourself professionally. A person may be an idiot online, but anyone you feel strongly enough about to attack is probably someone you have some kind of respect for, no matter how begrudgingly. And seriously, life is hard enough.
Minimize the apologetics
You're making an argument. You believe in it. Don't spend 13 paragraphs qualifying your argument or presenting devil's advocate statements against yourself. You know how annoying it is when someone apologizes to you orally at such length that they're not apologizing at all, just trying to get out of trouble? Perhaps you've been that person yourself. Don't be that person in print, which is even more tedious. Succinctly present your respect for the person you're attacking and get on with it.
Name who you're attacking
There are quite a few writers I love and respect otherwise who do this thing where they issue a half-assed formulation ("some people inexplicably believe" or some such) which allows them to attack without attacking. I guess the idea is to avoid personal conflicts (which works unless the person you're attacking reads your piece and gets offended nonetheless) and to not mess up someone's Google. The latter is creditable, but ultimately this kind of pussyfooting is annoying: if the reader doesn't know who you're referring to (which is eminently possible), they'll feel vaguely annoyed and spend hours googling the things you've paraphrased for the express purpose of being un-Googleable.
Link to the person you're attacking
This is basic courtesy both to the reader (who deserves a chance to independently evaluate the thing you're attacking in its original context) and to the person you're attacking (because at least they can get some traffic out of this). This is why asshole conservative websites like Breitbart.com only link to other asshole conservative websites (like "Newsbusters" which come the fuck on): they want to deny traffic to someone while attacking them, i.e. garner non-reciprocal attention. This is rude.
Be civil and email the person you're attacking
This is not a "writing" step, but it's a good idea. There's a good chance that the person whose argument you're going after may not even be someone who you have much in common with POV-wise. But they're a fellow writer, which is a hard profession, and unless they're the worst person in the entire world, you probably don't wish ill upon them. Remember, too, that a certain amount of online conflict is WWE: posturing for attention, fighting without ill will. We're all in it together. E.g.: I do not have the same taste as Dan Kois, to put it mildly, but I emailed him to let him know I'd be attacking him, and he was a total mensch about it, even linked back to me in a follow-up New York Times piece. Unless you exist in an echo chamber solely comprised of the like-minded, it behooves you (both personally and professionally) to accept that you'll be interacting regularly with people who are too low-/high-brow for your taste. That means you won't agree on much, but that's no reason to write them off in toto. If someone or their argument gets under your skin enough to go to the trouble of writing against them, that must mean you feel there's something there worth responding to at length. Otherwise you'd just go on Twitter and be a dick about it. So knock it off and give them a heads-up.
Don't be Kevin Smith
In the movie Chasing Amy, incredibly truthful moments are defused by really lame jokes, just to make sure no one gets too uncomfortable. When you're going negative, resist the temptation to. Write like this? Because like you're like a funny and decent and cool person? And like you don't know what you're saying and you feel insecure and WELP oh no brb? You're going negative and you believe in what you're saying; don't leaven what you're saying with the stupid stylistic tics of momentary internet trends. Have the guts to pursue your argument without the usual JUST JOKING I'M NOT JOKING JUST JOKING JKLOL scaffolding.
Invective is one of the cheapest forms of entertainment. Don't keep stopping to make jokes, but try to entertain. It leavens the anger.
Do it once, then move on with your life
Assemble all your points into one blast. You may forget something when you're writing it. That's OK, you can qualify and adduce in the comments. Don't go on and on forever in multiple dispatches about it. That's tiresome.
Bonus optional variant: Secret Cambodian bombings
Sometimes, in this life, you run into people who slander you and are just complete fucking dickheads. They make things up about you and don't care about anything. There is no reasoning with such people, and it's really better not to even try to do anything about them. But if you must, unqualified, hyperbolic invective is acceptable. Be careful though: make sure you know exactly what you're writing and that it's correct before you do it. There are no second takes on the internet, and there's always someone with a longer memory than you. Be careful out there.