The alligator clip on the end is meant to clip on to some type of grounded metal (such as galvanized steel conduit), and you can remove it to stick the plug into the ground plug hole on your wall outlet. Disregard the Voltage markings on the drawing (they are not quite accurate anyway).

This is very dangerous, you should never plug anything into a wall that isn't intended to plugged into the wall. Stray and unbalanced currents and voltages use the grounding to balance load and can be energized. All that is needed is equalizing your potential to the part by touching the part ground or by touching something that has been properly grounded, like an appliance with bare metal that is accessible. Grounding is important but shouldn't be treated as if there is no potential on the conductor, if the grounding conductor is energize it can hurt you or your stuff. I have personally tested grounding conductors with 60-70 volts on them in peoples homes. NEVER plug things into a wall outlets that weren't designed to be plugged in or UL tested.

Any tips would be greatly appreciated. I'm gonna drop about 1,100 on parts.

This is some of the best advice on here. Static discharge will ruin your computer and you won't even notice. It takes upwards of 2000 volts just to feel the discharge and around 3000 just to see it and 50 volts can seriously damage components.

If your house was built to code, you should be 100% safe in doing this... that is the point of that plug. That plug is connected directly to the ground (i.e. the galvanized steel conduit which is required to be true-earth ground). If this were not 100% safe, it would be a horrible idea to even touch the outside of your computer case... or any case that is metal, in any electronic component that has that third plug.

Seriously, if you have ever been in a house that had 60-70 volts on that grounding hole I hope you called the City, because it would not be long before that house will burn down. This is not up to code, and you run SERIOUS risk plugging anything into any outlet in a situation like that.

Point in fact, when you use the alligator clip to chassis method, you are supposed to plug your case into the wall (while leaving the switch off). This is effectively grounding the entire chassis and your wrist strap to that third plug in your electrical outlet.

The only time I would be concerned is if your house is much older construction (world war 2 era) and you had some 220 V outlets in your house. US electical codes were a little bit more forgiving then, and it is possible that the ground plug was used a neutral while the other two plugs were out of phase 110V lines.

Of course, if that is the case start suing everyone and anyone that was involved with your house sale.

Never place your life in the hands of anyone. Attaching yourself to a circuit is always a bad idea. Yes, up to code the circuit should be doing its job but if something is wrong then you are dead or injured. I have seen too many mistakes in new electrical construction and installations to absolutely trust ANY circuit with my health. I have a degree in electronics and I am an industrial electrician and I can tell you that things mess up, get broken or installed improperly. The code is very powerful and a great tool but too many things can go wrong and we should never trust that one conductor is going to save us. Grounding conductors are emergency equipment and should be treated as such and by attaching yourself to a device that is made for emergency, worst case situations you are asking for trouble.

The reason I have found voltage on grounding conductors is because I was there to fix the electrical problem. I have found voltage in new construction, old construction and everything in between and they are usually caused by people. You wouldn't believe some of the things I have seen in homes and in industrial plants.