Okay, there's a bit of a difference between subtlety and fridge brilliance. I think you're confusing the two and I need to make it clear they aren't the same thing.
Subtlety is hinting at something, generally relevant to the overarching plot, that readers may not get the first time. It's done by all manner of things, like (good) foreshadowing, slow and well-measured character development, the seamless incorporation of plot objects, etc. Subtlety is usually key in executing subplots and, by the end of the story, the things you were showing should be clear to the reader, even if they aren't entirely sure why its clear to them. Basically, you're taking tons of little tiny threads over the course of the story and weaving them together so that when all the cards are on the table, suddenly all these little things make sense, but it'll take the audience a second read-through for them to understand why they had that "Eureka!" moment in the first place.
A good example of this would be going back and rereading the Harry Potter books after finishing the seventh, and looking for all the little instances where Snape's behavior towards Harry seems out of character from the brusque, sardonic and surly authoritative figure he usually is. In a rare case, this is also quite evident in the movie adaptations, because Rowling took care to tell Alan Rickman how Snape's character ended up in the end in the early days of filming Sorcerer's Stone. So all along, Rickman knew exactly how Snape would react to any given event as it concerns Harry given his history with Lily and Dumbledore. There are cuts in scenes of him (in the early movies most especially) looking incredibly concerned or worried when it would be far more in-character (or so we'd think) for him to be smug. That's not fridge brilliance, that's just incredibly subtle little things that didn't make sense until everything came to light in the end.
Another (different) example of this would be a novella I read for class early this year, The Nun of That. I can't remember the writer, and I'll edit this post if I can. Within the first two pages, the narrator says that her daughter and her best friend are imaginary, but she's kind of a strange woman who does some strange things, and since the daughter and friend are written as convincing characters, you sort of disregard that they're imaginary. Little clues over the course of the story remind you that, yes, they are, but you as a reader want to think that it's just the narrator being weird. But in the end you realize that they never existed and she's built up this whole fantasy life for herself, and while she was totally up front about it it was you who convinced yourself she was lying.
Fridge Brilliance is a bit different, and it tends to have its roots more in symbolism than subtext. Not to say it can't be subtext, but it's far more common to see it in symbols and actions employed in the work that convey a psychological resonance. You cited Korra as an example, so I'll use that in my examples. Korra is meant to be Aang's opposite, and in many ways, he overall demeanor reflects that. There's also the fact that the events of the first season take place in the fall, while the earlier series took place in winter, then spring, then summer. It picked up right where the last one left off, albeit 70 years later. Then there's Amon's use of the red disk, which pretty universally means the sun. He was supposed to be looked at as an enlightener, a bringer of a new age of reason. Not to mention, he was the Avatar the city needed, one who could have brought balance and harmony except that he went about it wrong. For my last example, I'll call up one of the last scenes of the finale, with Korra on the cliff. One could assume that yes, she just ran as far as she could to be alone and to think, but there's also the possibility she meant to commit suicide there. Certainly, standing on the edge seems to poke at that.
If you look through other TV Tropes Fridge Brilliance pages, you're going to find that a lot of the examples in there are symbolism, and that in turn conveys a subtext, because we are psychologically wired to associate certain things with other certain things, and it's only when we go back and look at them understanding that that we begin to understand why we associated those two things together in the first place. Does that make sense? I hope it does...
If you're looking to get more into a character's psychosis, you'll want to look more at subtlety and subtext, and less at the potential for fridge brilliance. To do that, to have people go back and say "Oh, that all makes sense now!" or "So that's why I associated this with that", you'll have to understand people first. The best way to write convincing characters and all that other stuff I've talking about is to watch and listen to people doing what they do all the time. Someone could say "I'm fine", but what they could mean is "I'm in desperate need of a hug" and it's our job as writers to convey that through our writing. So watch people as they pass on the street, at how they carry themselves and form their words and use their intonations. Watch movies with skilled actors who understand emotion and how to convey it believably. Read Chekhov and Shakespeare and Dickens and Shelley (Mary, principally, but I'm sure Percy did something right somewhere) and Twain and Hugo. You can't write psychological depth until you understand psychological depth, and writing with subtext takes a lot of practice. The best thing to do is try it, and when it doesn't work, try it again, rinse and repeat.
