The AR-15 bolt/carrier assembly is in many ways the most critical portion of the AR-15 weapons system. It is involved in the duties of feeding, firing, and extracting ammunition in the rifle. Though much maligned for its direct gas impingement mechanism, which tends to result in a dirtier action, the design does incorporate some self-cleaning attributes to maintain reliability. These mechanisms alone, though effective, are designed more for stressful field conditions when regular maintainance might not be possible. Under the conditions most civilian shooters see, it's wiser to maintain a procedure for cleaning, rather than risk lessening the lifetimes of these expensive arms..
A consideration to make when cleaning firearms is that there's no need to be overzealous about grime; there's no drill sergeant barking over you. In fact, the better part of firearms wear is due to overcleaning. Ex-military types are especially prone to this; one has to keep in mind that firearms maintainance skills acquired in basic training are in large part a method to instill discipline.
Disassembly
We start with the bolt/carrier assembly, taken from the upper receiver. It's usually covered in all sorts of soot and dirtied oil, so a rag or paper towel is used to wipe off the most offensive areas.
We flip the bolt so that the left side faces us ("left" being relative to its position when viewed from the weapon firer's perspective). Using a fingernail or some other implement (screwdriver, in this case), we remove the firing pin retention pin and set it aside.
Now that the firing pin is no longer captured, it can be removed from the assembly
The cam pin is then removed; this involves first pushing the bolt into the carrier assembly, as illustrated, then rotating the cam pin from its "horizontal" orientation to it more "vertical" orientation. You can't lift it out if you've got it wrong, though, so experiment if you have to. It helps to use the firing pin to rotate the cam pin if you've got gloves on, or are less dextrous (fat fingered!)..
Once the cam pin is removed, the bolt can be pulled forward of the carrier.
At this point we've disassembled the bolt assembly as far as is normally required for regular maintainance... but if we want to go farther, disassembling the bolt isn't too hard...
Using the firing pin, we push out the pin retaining the extractor. This can be done from either side.
The bolt is now disassembled as far as you should ever need to. Any farther, and you'll need more tools. Re-assembly of the bolt and carrier assembly is exactly the reverse. Make sure to pull the bolt forward before you insert the assembly in your upper, though, otherwise the cam pin will interfere.
Analysis
Within the bolt, it's interesting to note a point of failure in the AR-15 system. The extractor springs in most AR15 bolt assemblies are fairly weak, and this can lead to various extraction-related failures. One of the few high points about Colt assemblies is their usage of higher strength extractor springs. You can tell the difference by looking at the inner plastic insert. "Normal" springs feature a blue plastic insert, Colt strong springs have a black insert.
Owners of low power extractors need not despair, though.. salvation can be found in purchasing higher power assemblies from Colt suppliers, or Wolff Gunsprings. If you're a poor college kid like me, though, you've probably seen the blue rubber "D-fender" (example link) and wondered just what's so special about that "polymer" of theirs. Well, "nothing".. Just go to a hardware store with your extractor and pick out an o-ring that fits over the spring. I forget exactly what size it is, but it's a good fit, and improved extracting in my previously ailing carbine..
The o-ring installed on the weak spring, alongside the Colt extractor..
Last thing to consider.. Even with the best of precautions, failure can always occcur! A spare carrier assembly could salvage a bad range day, or maybe even save your life. Most military types like carrying at least a spare bolt. At very least, tape a spare extractor to the inside of your pistol grip; it'll stay out of the way, and still be there if you need it.