I want to be clear about something before I describe the right way to elbow pick. First of all, it is NOT a replacement or alternative to picking from the wrist or fingers. It sounds slightly different, and can allow you to pick just a little bit faster, however, this doesn't mean you should start picking from the elbow. The reality is that if you have good wrist picking technique, you will be able to pick from the elbow correctly, and the more you practice wrist picking, the easier elbow picking will be when you want to use it occasionally. Proper wrist picking has more to do with position that anything else. You want the pick to travel straight down and up. If it's not moving straight up and down, you need to adjust the position of your wrist. Lay your hand completely flat on a table and move it from side to side. Find the location along this path where, when you twist your wrist, the pick goes straight up and down. This is the spot your wrist should stay in when you're picking.
So what is the wrong way? Well, the elbow is just too far away from the strings to pick quickly without tensing up. It requires some form of stabilization. Bad guitarists do this by tensing up the muscle behind their elbow and 'spazzing' their elbow like they're playing an arcade game. This is bad because it doesn't allow you to pick individual notes with distinction, you're unable to move strings fluidly, and if you do it for even a few minutes every day, it will result in real injury to the tendons in the elbow.
So, if that is the wrong way, what's the right way? Again, elbow picking requires some form of stabilization. The best way to do this is to anchor the bottom corner of your palm to the guitar. You can press your wrist toward the guitar to keep it in place, but you don't need to press too hard. The difference between this and the former method is that this method only requires one group of muscles, not too opposing groups of muscles both pulling on the same joint, which is what leads to injury. Here's how you know you're doing it correctly: When you move your elbow down, the anchor point will make the wrist rotate slightly, however, it rotates outward, instead of inward. What this means is that, for a downstroke, your wrist will twist outward, instead of inward. (This motion alone is very good. Practicing this motion itself can greatly increase your speed and accuracy, regardless of how you choose to pick when you're really playing. The motion for a downstroke can be identified like this - point your arm straight out, now rotate your wrist so your thumb goes over your wrist. This is the opposite of what most guitarists do to perform a down stroke, however, it is superior for certain types of single note runs, and great for sweeping. It's best to be able to move between this type of picking, normal picking, and elbow picking). This allows you to elbow pick with stability and accuracy, and allows you to make string changes with by shifting your wrist. If you do it correctly, it will allow you to pick faster than you normally would, but with the same level of control. Your arm will get TIRED, but it won't get injured. It should feel like you just used a 'shake weight', but not as if you're elbow is being torn apart.
Again, don't elbow pick just because you can't wrist pick. Learn to wrist pick, and when you can pick cleanly at shred speeds (200bpm or more) then you know you've got it down, and it's okay to use elbow picking sometimes without worrying about injury.