First, there's a common misconception that I want to address before I even begin. I've heard way too many people try to claim that they don't write for an audience or that they only write for themselves. In my mind, this usually translates to something like, "You or someone else gave me a critique I don't agree with, so I'm trying to justify why I'm going to ignore it." You're going to have a hard time convincing me that you don't care about anyone else's opinion of your work if you PUBLICALLY SUBMIT IT ONLINE.
I don't know if you've noticed, but dA (and any other site like it) is essentially structured to be used for peer review. That's the main point of the ability to leave comments in the first place. If you're really only writing for yourself, you would keep your stories in a shoe-box hidden under your bed. And, no, the "I was posting it so my very bestest friend Mary Sue could read it" excuse doesn't fly either. That's why e-mail was invented.
So, if you're still with me so far, I'm going to assume that you agree that, yes, you care about what John Q. Public thinks about your work. Also, since the title didn't scare you away, you can freely admit to yourself that there is always room for improvement in your writing and that your peers can help you get there. Congratulations. You have already taken the first (and second hardest) step in becoming a better writer. At least 90% of the people who would consider themselves writers don't get past that step. Pat yourself on the back and we'll move forward.
1) Finding a Critic
This can understandably be a daunting process. Luckily, there are a few places on dA that are there to help you out with this kind of thing. If you're on dAmn (deviantART messaging network a.k.a. the chat rooms), try looking in #getLIT. There are usually a fair number of active people in there, several of whom are typically very willing to give a critique. If you'd rather try the forums, you'll want to check out the Lit Workshop Forum (just be sure to read the forum rules first!). Finally, there are a few club accounts out there that offer critiques. LITplease is one example that is still active.
2) Picking a Critic and Approaching Them
This will really depend on where you found your group of critics. If you're in a chat room, you can address the entire room and hope someone is free enough to critique, or, ideally, you'll do a little research first. Try to find some examples of how a particular person critiques (hover over the stats on their profile until you find the one that says recent activity, then click more activity to see their recent comments) to find someone you like. Different people will have different styles of critiquing. If you're going through the forums or a group site, you probably won't be able to pick and choose your critics. On the other hand, you'll probably get the benefit of having several critics working on the same piece! Just follow the rules and guidelines of wherever you are, and you should be fine.
Now, if you are in a chat room, then you're going to be a bit more face to face with your critic, so there are a few points of etiquette to follow:
A) Ask politely. No one's going to pay you any attention if you're telling them to critique your work. Remember that they're doing you a favor and not the other way around.
B) Proofread your work BEFORE you ask for a critique. Essentially, that means make sure you're using proper spelling and grammar. If you're not even going to commit yourself that much to fixing your writing, you'll just be wasting the critic's time. Proper spelling and grammar should go without saying.
C) Be sure you use proper spelling and grammar when you talk with your critic. Think of it as a job interview. It's hard to take someone seriously when they ask you to "plz raed there storyz," even if their actual writing is good. I realize that it's a chat room and it's called chat speak for a reason, but just take a second to consider how it reflects on your writing ability.
D) Only ask for critique on one piece at a time. A good critique takes time. Your critic isn't just reading it; they're going over your writing with a fine tooth comb. Plus, they're spending their free time giving you a helpful service free of charge. Don't get greedy.
3) What to Expect From Your Critique
Alright, so someone's giving you a critique. Awesome! First thing's first, be patient. Like I mentioned earlier, a good critique takes time. Don't ask for a critique unless you're not busy and don't have anywhere to be, and, please, never nag a critic to hurry. Now, remember how I mentioned that realizing you need critique was only the second hardest step? That's because the hardest step is accepting the critique.
The best (and, of course, hardest) thing to do is to simply remove all emotional ties to your piece after you've finished writing it and before you submit it to be critiqued. Keep in mind that your piece of writing might very well be slashed to ribbons and tied back together in such a way that you don't even recognize it anymore. Sometimes, it's necessary. Just remember that it's nothing personal.
In fact, that needs to be repeated: IT'S NOTHING PERSONAL. Unless you're dealing with a terrible critic, they aren't going to be personally insulting you as a writer, even if it may feel that way. And there you go: part of the hardest part about accepting a critique is to realize that being told a particular piece was written poorly doesn't mean they're saying that you're a bad writer. Even the best writers will write crap from time to time.
Now, that doesn't mean you have to agree with everything a critic has to say. Just be willing to admit when they're right. Even if your critic tells you something you don't agree with (which will happen a lot), try to consider the fact that they may be right. Even if their suggestion feels off, it could help by drawing attention to something that could be changed in a completely different way than how they suggested.
Depending on who is giving the critique and how, you may be able to engage in a dialogue with your critic after they finish or even during it. A lot of that depends on the critic though. Some like to give their critique to a patient, listening audience while others prefer an open dialogue. If you're not sure what your critic wants, politely ask if they mind discussing it with you point by point. Again, disagreements are common, just don't let it become an argument. Also, don't forget to thank your critic at the end for taking the time to help you!
When all is said and done, the most important thing is make sure you learn from your critique. That's the whole point! Even if you only take one thing to heart from the critique, then it wasn't a waste. Good luck and good writing!