I do something that's probably quite a bit different from most other engineers. I sort of approach this like others might mic a floor tom. I want to get as much of the tone and character of the kick as possible, and I believe this simple method works.
My starting position is to put a large diaphragm condenser near the front of the kick drum, not directly in front of the kick as others often do, but rather, looking at the rim. It's slightly off to the side, aimed more or less at the rim between the skin and the side of the kick, about 4 inches away from the rim.
From there, I move it around, depending on the song, to get either more of the shell or more of the skin, simply moving the angle of the mic to get more or less of the shell/skin, or closer/farther to get more impact as needed. It's very simple to move the mic around while wearing headphones. I find that this way, I'm able to record the shell/character of the kick drum as well as the impact, achieving a nicely balanced, natural sound with just one mic.
If there is too much bleed from the rest of the kit, I will place a couple of wooden chair wrapped with acoustic foam taped on to it over my kick mic. If that doesn't work, I also have a hollowed-out garbage can with foam in the interior, but this is very large and difficult to walk around in my tiny little living room, so I prefer the chair. You must remember, I am the Mayor of BudgetLand, so whatever works.

Processing: Nothing while recording. Compression and EQing later.
On these examples, I used a AT4060 tube condenser through an FMR RNP mic preamp. Neither of these were even necessarily ideal choices (although you could do far worse than these!); I simply worked with what I had at the time.
Here's a link to some songs that show how this technique sounds:
http://www.elevenshadows.com/studio/index-studio.htmGo to the blueberry-colored MP3 player and play any one of the following:
"Collage of Songs recorded at Blueberry Buddha" (anything with drums) or "Hang On" and "Undone" by Nectarphonic. All of these are fully intact drum performances with no replacement.