What do I need to do for the 72 hour Visa?

Amanda, where will you fly to after you arrive in Beijng? If you fly back to South Korea, then you won't be able to make use of 72 hours free transit. If you fly to another country, for example, getting back to USA, then you can apply for 72 hours free transit.

In this case, your USA passport and onward air ticket (confirmed seat) to USA are required. You needn't make any paperwork in advance.

I will fly from Tokyo to Seoul, Seoul to Beijing, then Beijing back to Tokyo, where I live. Would this flight arrangement work for the 72 hour Visa? Besides showing proof of my next flight do I need to complete any forms?

Yes, the flight arrangement works for the 72 hour free transit. No forms are to be completed, you just need to fill in the Entry Registration Card.