Believe it or not, the economics of the chop shop are best demonstrated in the tech world. If you're sitting on a stack of non-working, half-disassembled computers and peripherals, you'll make more money selling them part by part than you would trying to make a few working computers out of them. While there is money to be made by recycling the valuable components, pulling the parts from the computer and selling the specific parts to users who need that exact part can be a windfall.
Take inventory of what you have. Note the make, model and speed of the part. While some computer components are compatible with a wide range of computers (a third-party video card, for example), some parts are specific to that brand or even exact model of computer.
Disassemble the computer and store the parts, labelling them correctly. Use static bags to store any components that are sensitive to static (RAM, motherboards, PCI cards) to keep them protected. If you want to wipe the data off of any hard drives, do that at this time. You can delete the files by putting the hard drive into another computer, or an external hard drive case, and reformatting it.
Put the parts up for auction on eBay or a site that specialises in computer equipment. You'll find that the market for a specific part is pretty strong, because it's often one simple component that stands between having a working computer and forking over the dough to buy a new one. Put as much specific information you can in the listing, along with a picture. If you need more data on a part, or you're unsure about something, do an online search to find out the specs.
Sell the valuable parts (the gold from the various components is the only real valuable metal there) to a metal refiner. These are companies that specialise in removing the gold from the circuit boards. The rule of thumb is that the older the computer, the more value it has in this department, as newer models employ less valuable metals. Again, a simple online search will yield scores of companies that specialise in this.
Recycle the carcase of the PC in a way that is environmentally safe. Just putting it in a box and having the dustman remove it isn't the correct way to rid you of the unusable parts. Do an online search for computer recyclers in your area, and make a run there with as many computers you can, once you've stripped out the good stuff.
Offer a "no questions asked" refund if a buyer claims a part doesn't work.
The inner workings of computers can be razor sharp, so take care when removing them.