Actually, I think that this is much less about the technical settings and the camera equivalent of the human eye — which, by the way, is covered in this earlier question, and this one as well — and more about some actually somewhat heady questions of identity and self-perception, as well as a more mundane but still philosophical issue of the difference between fluid motion in time and a frozen frame.
There's some degree where, if you want a certain look, you'll need to replicate the lighting and technical factors that match your mirror setup. You'll probably want to start with a lens with a "normal" focal length — that is, a little less than 50mm-equivalent — somewhere around 28-35mm on an APS-C camera.
But, here's an important thing. Your mirror provides a 1:1 representation. Are you going to print at that size? Maybe if we're talking about a hand mirror, the answer is yes, but if not, suddenly you're working with a very different perception of "normal". That means, you might actually get better-feeling results with a longer focal length. (In general, I think printing so that you are life-size is an important aspect of getting a true mirror feel, if that's an important part of this project; even if you might end up cropping very heavily as if looking in a small mirror.)
Beyond that, though, is one of the intrinsic issues in all of photography. When you're looking at real time — even if you're holding very still — little awkwardnesses and imperfections and so on appear less so than they can when captured in a still. The flicker of a smile which moves from the lips to the eyes and back can be powerful and subtle in real life — and totally missed in a photo. In fact, it can look really weird if caught in the wrong instant. When you're looking in the mirror in real life, millions of years of human development in recognizing these subtle cues is at work, and it goes directly to special centers in the brain dedicated to figuring it all out. With a photograph, you're cutting all of that out.
The best way to learn about this is to practice — the self-portrait skills here aren't particularly different from those in capturing expressions in portraits in general. Arranging your own expression in a mirror is something different, though, and I think the only answer really can be "practice, practice, practice". One important aspect, of course, is eye contact — you can't see yourself in a mirror without it, so make sure you're looking directly at the lens.
And of course, there's the issue that mirror-image people look slightly odd and different — if your friends were kidnapped in the night by aliens and returned inverted, you'd notice that something was off, even if you couldn't pin it down. But it's beyond that. If you stand next to a friend and look in a mirror, they'll see you very differently from the way you do, not because of the angle, but because our impression of human beings is complicated and not a matter of optics.
Vincent van Gogh painted dozens of self-portraits over his lifetime. I think it's safe to say that these represent his perception of himself in the mirror much more powerfully than a painting or photographic portrait someone else made ever could. I think these are probably very true to what he saw in the mirror, even if they are not always completely literal.
So, then, there's really a question. Do you want to do that — capture your inner perception of yourself for all to see? Or, is it really more hey, I kind of like what I see in the mirror, but I can't get it in a photograph? Know which of these you want to aim for will help a lot, but I think in either case, once you have a basic setup, the approach is to take a short series and immediately review the results (ideally, on a big screen, not just the camera's LCD — see the point above about the size of a mirror).
Examine the technical, and make adjustments to better match your desired results — watch the lighting and shadows particularly, and decide if the angle is right for your perception of your features in the mirror. Play around with size and cropping.
Most importantly though, watch your expression and the minutiae of minor differences in your posing. Does it look too stiff? Not angled right? Go back right away and do another series with those changes, and repeat. (This isn't a magic bullet solution, but there rarely is!)
The idea of using a half-mirror is an interesting one, and may help you get more direct results. Be aware, though, that the existence of one-way mirrors is a myth. There's just glass that's half reflective, and if you're in a well-lit room and the other side is dark, it's a mirror — while if you're in the dark room, you can see through into the light. It might help better with eye contact than the other approach of putting a mirror next to the camera and shifting slightly to look into the lens. I'd be interested to see your results with that approach.