For the freshest possible product (barring catching the crab yourself), you would buy a live crab and steam it yourself. If you do buy a fresh crab, don't boil it. Boiling it will cause all the flavors to leech out into the water. What a waste.
Unfortunately, the seafood vendors at Pike Place do not sell live crab. It was either Dungeness crab meat for $40/lb or whole cooked crab for $10/lb. Being the cost-obsessed, poor student that I am, I figured with the average whole crab at the market being 2 pounds or $20, I would have to get at least 8 ounces of crab meat to be worth the trouble. There is the added benefit that a whole crab comes with the flavorful liver, which is I wanted for my recipe, and the meat might be fresher than the meat in the case. So we bought the whole crab and hoped it was worth it.
If you have a live crab, steam it for about 6 - 7 minutes per pound. When you place the crab in the steamer, it will undoubtedly try to escape. Show no mercy and hold down the lid until it struggles no more.
With your purchased cooked crab or steamed live crab, run it under some cold water and scrub the exterior with a stiff brush, especially the belly and legs.
On the belly side, there's a flap of shell, which I believe is called the apron (long on male crabs, wide on female crabs). Lift it up and break it off close to the body. This will leave you with a hole that you can stick your finger in to pry the shell off the body. If it's a freshly steamed crab, there will be more fresh juices so make sure to do this over a bowl to save all of the flavorful juice. Hold the crab with the top shell down, which serves as a vessel to hold the juice. 
Set aside the body. Inside the shell, there will be some yellowish-green stuff. That's the liver or tomalley (lobsters have that stuff too). The whitish pieces are albumin, which is harmless protein and fine to eat. There will be some thin transparent membranes, not tasty, and some membranes covered in purplish stuff, also not tasty. Take a spoon, scoop out the liver and albumin into a small bowl and set aside. If this stuff disturbs you, you don't have to eat it. I think it makes a lovely soup or sauce and I try to use it in my recipes. The color of the tomalley is more appealing in a freshly steamed crab. It's more of a muddy green in an already steamed crab so I don't blame anyone by being grossed out. Female crab, not commercially available, have a delicious bright orange roe. 
After you scraped out what you want from the shell, discard the random membranes and the purple stuff. If you want to keep the shell for serving, break off the mouth pieces and random crab bits here and there, rinse it out, and set aside.
Now moving on to the body. There will be some residual liver, scrape it into your bowl if you'd like. I didn't get a good picture of this but there will be multiple white triangular shaped, feathery, spongy type things lining both sides of the body. Those are the gills and are not edible. Peel them off and discard them.
Now you're left with the cleaned crab body. Hold firmly with both hands and break it in half where the body naturally divides. Crack the crab such that you bring the exterior of the shell together (let me know if that doesn't make sense). 
Now you have two crab halves. With each crab segment, pull the legs from the body like breaking apart a chicken wishbone. You want to pull them apart such that the break will sometimes separate the crab's cartilaginous body. Sometimes, you're not so lucky and the legs break off cleanly. Once you have all 10 legs separated, if any of them are connected to parts of the crab body, separate the body from the legs so you have 10 clean legs and various bits of pieces of the crab body. When you break apart the crab, try to break and pull at the same time. Sometimes big nuggets of meat will be pulled out of the shell with a break, making your job easier.
Now, you can start picking the meat meticulously out of all the nooks and crannies. A long pointy object like a chopstick is very useful for pushing the meat out. Do things in systematic batches. First get the meat out from the body pieces. The body meat is delicate and sweet but encased in a thin cartilaginous maze. When working with crab, you want to use a delicate hand so you don't mangle the meat. Try not to break up the meat too much. Sometimes you can peel the shell apart from the meat, other times you need the chopstick to push the meat through.
When it comes to the legs, you can use a crab cracker or a swift whack with a mallet. For the smaller, delicate tips of the legs, you can sometimes cut the shell with a pair of scissors and fetch the meat out that way. there's sometimes a center sheet of cartilage that runs through the meat, you'll have to make sure to get that out. 
From the start of washing the crab, to weighing all of the meat, it took a total of 27 minutes. The 2 pound crab yielded about 12 ounces of meat and about 2 tablespoons of tomalley.
