The Illustrated Guide to Bearing Cleaning

A dirty bearing To tell if your bearings need cleaning, check:  Do they make noise?  Are they running slower than they used to?  Do they squeak, scrape, or just sound funny?  Are they dirty on the outside?  (Chances are, if they're dirty on the outside, they're dirty inside.)  Skating through dirty or sandy areas, or in the rain, can really get dirt inside your bearings.

In this picture, you can see dirt on the shield where it meets the edges of the bearing. That probably means there's dirt inside.
An allen wrench On most skates, you can remove the wheels by using an Allen wrench.  My skates take a 4 millimeter Allen wrench.  Some skates require you to use two allen wrenches, or do other funny things to get the wheels off.
Using a skate tool to get the bearings out To get the bearings out of the wheel, it's helpful to use a skate tool, or using a round object the size of the inner spacer.  In a pinch, a screwdriver will work, but it's less convenient. My skates have an inner spacer just slightly smaller than the inside of the wheel, which means that the spacer can move around a bit.
Spinning the bearing to see if it's dirty You can spin the bearing with your fingers to determine how dirty it is or how badly it needs cleaning. Most bearings have a shield (the gold part) on both sides.  Some bearings only have a shield on one side. 

With most bearings, the shield is held in by a C-clip that goes almost all the way around the shield, and it fits into a  little groove in the bearing race. On less expensive bearings, the shield may not be held in with a C-clip, but instead just pressed in. In this case, the shield can be pried off with a sharp object, but you won't be able to reuse the shield.
Using a pin to remove a C-clip The C-clip can be removed by inserting a sharp object underneath the edge of the clip.  It's actually necessary to do this with two hands, but I had to hold the camera with one hand. Try to keep the C-clip from flying away while doing this, and don't poke yourself! (You can also use a dental pick to do this. Ask your dentist if he/she has any old picks being thrown out.)

By the way, the wheel in the background is an example of a worn wheel that should be replaced.
A disassembled bearing The bearing has its C-clip and shield removed.  If you have serviceable bearings with C-clips, you can remove both shields if you're really meticulous about cleaning, although it's only necessary to remove one shield so the solvent can get in there. On the other hand, if you have non-serviceable bearings and you have pried off one shield, DON'T pry off the shield on the other side!
A number of bearings removed from the wheels Here are the rest of the bearings after being removed.  They're pretty dirty, although they weren't really scratchy or slow or anything.  I'm cleaning them mainly for the outside dirt.  It also appears that they over-lubricated the bearings when first building them, since there was grease on some of the inside bearing surfaces.  This extra grease probably attracted some of the dirt.
Citrus cleaner/degreaser kit To clean the bearings, you will need some kind of biodegradeable cleaner or solvent.  I'm using a commercially available bearing cleaning kit.  Citrus-based solvents seem to work well for this purpose, and you can often find these bearing cleaning kits at sporting goods stores. I've also used Finish Line degreaser (in the back) with good results.  Best of all, it's the same stuff I use to clean my mountain bike chain.

There are a number of different bearing cleaning kits available, and the choice is largely a matter of personal preference. If you ask a group of inline skaters for their preferences, you'll get a number of different answers. One thing to avoid, however, is a flammable solvent like gasoline or lacquer thinner. You don't want to blow up your home just to get your bearings clean.
Pouring solvent on the bearings Place the bearings into some kind of pan or something else that can hold them.  This cup came with the kit, but you can also use a shallow pan. Fill the pan or cup with enough solvent to cover the bearings. Shake or agitate the bearings to make sure the solvent gets to all those sensitive parts that need it.
Bearings removed from the solution and put out to dry Remove the bearings from the solution. Some cleaning techniques and solutions recommend washing out the cleaner with water. If you do this, you should make sure to get all the water out of the bearings before relubricating them, so you don't end up with rust. Put the bearings on a paper towel, open side down, so they'll dry.
Spraying the bearings with a spray duster

If you're impatient, you can spray the bearings with a spray duster (like those used for computers) to dry them out faster.  Actually, when I did this, some of the old grease came out of there, so it apparently wasn't completely removed by the citrus cleaner.  Note that I'm not actually pressing the spray button because I needed a hand to operate the camera.

At this point, I let my bearings dry overnight.  This is to make sure they're all dry, so the solvent won't do bad things to the lubricant.

Lubricating the bearings Once your bearings are dry, spin them to remove any extra cleaner, and lubricate them with an appropriate lubricant.  Basically, most people use either some kind of grease or oil to lubricate their bearings. Grease works well to keep things lubricated, and it can hold some stray dirt out of the way in the bearings. On the other hand, many people prefer oil because it helps the bearings run faster than grease. If you're going to use oil, don't use 3-In-1 or sewing machine oil -- they aren't really formulated for inline skate bearings, and they could gum up after a while. You should be able to find either grease or oil at any sporting goods shop.

I'm using Castrol Professional Race Oil (since it protects against viscosity and thermal breakdown). It's important to avoid over-lubricating the bearings, since this will slow the bearings down and tend to attract more dirt. Spin the bearings to make sure the lubricant is evenly distributed.

It's up to you whether you want to replace the shields or not.  Some people don't replace the shields and just point the now-unshielded side toward the inside of the skate.  I prefer to replace my shields for extra protection.  Place the shield back on the bearing and insert the C-clip into its little slot in the bearing race.  Again, spin the bearing to make sure everything is in working order.
Diagram of wheel rotation I took this opportunity to rotate my wheels, since the right-side wheels were starting to wear a bit from T-stops. Basically, the idea with this 5-wheel rotation is to make sure that the front and back wheels (which get more wear) are moved to the middle, while the inside wheels (that aren't so worn) get moved to the outside. If you're on four-wheel skates, life is easier: exchange the front wheel with the third wheel, and exchange the back wheel with the second wheel (counting from the front).
Drain hose from the washing machine Wipe the dust off the inside of your frames, so it won't get back into your bearings.
Steps to reassembling bearings To reassemble the wheels, put one bearing in, followed by the spacer, and then the other bearing.  Make sure the bearings are inserted firmly, or they'll be hard to put back onto the skate.
The finished product: The skates reassembled When putting your wheels back on your skate, put the most worn surface toward the outside of the skates.  The inside edge of your wheels gets the most wear. Tighten the bolts enough so they won't loosen, but not so tight that the wheels can't spin freely. And you're done!  (Well, I am, anyway.)

Disclaimer: This is my preferred method for bearing cleaning, but there are people who would probably disagree with me on my choice of cleaning or lubricating products. It's kind of like picking a favorite laundry detergent: everyone has their favorite, but it's still a bad idea to wash your clothes in gasoline. For more information, check out The Bearing Maintenance File at skatefaq.com.