Taking a Spyderco knife apart is easy, but putting it back together can be a pain. This is the method I came up with after trial and error. I hate to make such a specific instructible, but I think these knives are popular and user-friendly enough to warrant it. In this instructible I assume the pocket clip will be in the position shown below, but it can be attached to either end of either side.
Step 1: Disassembly
First, take out the five main screws holding it together and the three holding the pocket clip on. All but the hinge are torx T-6; the hinge is T-9. Once you've taken off the clip and top plastic grip, gently pry the top metal plate off. The lock spring is under tension, when you remove it make sure it doesn't fly off and get lost. There are a total of 8 screws - 3x T-6 .2in screws, 4x T-6 .25in screws (3 for the pocket clip and one for the blade lock pivot) and 1x T-9 .2in hinge screw.
Step 2: Cleaning
Next, you can lift the blade off and set aside the bronze-phosphor washers above and below it. These are what needs to be cleaned if the knife won't open smoothly. Take out the lockback, and wipe any dirt off. Take out the plastic piece in the middle, the bottom plate, the screw posts, and clean everything.
Step 3: Cleaning the Washers
Take the washers one by one, and without touching them pinch them inside a folded piece of paper (>1000 grit sandpaper if you have it) and rub them between your thumb and index finger. This should clean any dirt off and leave them bright and shiny. If paper won't do it and you don't have extremely fine sandpaper, try softening up the dirt with rubbing alcohol on a Q-tip. Brasso on a rag would probably work too, but I've never tried it. Either way, make sure they're very shiny and dry before you continue.
Step 4: Cleaning the Hinge Surfaces
Once the washers are clean, clean the four surfaces they come into contact with on the blade and metal frame plates. These areas don't usually get very dirty, because the dirt tends to stick to the washers instead. Rubbing alcohol will probably be fine, but if they are really dirty then Brasso will do the job. These places should also be smooth, clean, shiny and dry before you put the knife back together.
Step 5: Reassembly
Take the bottom plastic piece, put the screw posts in it and put the bottom plate back on it. If you get the plates confused, the bottom one is the one with the shiny hinge surface facing up when you put it on the bottom half of the handle. Then, set a bronze washer on that shiny surface over the screw post (without touching the washer directly), set the blade on top of it, and the other washer on top of that.
Put the middle plastic piece in (the part that holds the lock spring) and put the lock spring in place. Remove the screw post on which the lockback hinges. Put the top metal plate on, along with the top grip, and replace the four screws (minus the lockback hinge screw and clip screws).
Step 6: The Tricky Part
The knife is now more or less assembled, minus the pocket clip and lock-back (all of the remaining screws are of the longer T-6 .25in type). Put the lockback in its slot in the back of the knife. Line up the hole in the lockback with the screw-hole. This will take some force, as the spring is resisting you. When you've got it lined up and held in place, push the screw post through the hole from underneath.
Remember, all of the screw posts have a flat edge that needs to be facing the "belly" of the knife. Doing things in this order is important because otherwise the tension in the spring will warp the plastic piece holding it and you won't be able to get the top metal plate to fit on top of it.
Step 7: Finishing Up
Now that the last screw post is in place, put the that screw in and reattach the pocket clip. You can tighten or loosen the larger hinge screw to make it easier or harder to open the blade.