Another one who doesn't fly. Cool. It's not just me!!
Shortest and cheapest are unlikely to be the same thing, however...
Best chance is freighter - a few companies offer the possibility providing you don't mind being classed as cargo. Standard way is normally Long Beach (or Vancouver/Seattle, but from CA Long Beach makes more sense) to somewhere like Incheon or Pusan in Sth Korea. From there ferry to China, and train down.
However, as you are going to Vietnam, look for Long Beach - Shanghai or prefably HK or even Singapore freighters. There should be at least one a week on each route. Somewhere like Haiphong is even possible, but guessing more expensive than a more main steam route.
You can go through agents (do a google search with variants on freighter/across Pacific/port names etc), and you get any number of places like this: www.freightercruises.com/asia-pacific.html.
Look around for best price, be flexible on departure, and if you can try phoning up/emailing companies individually and haggle, so as to avoid agent fees. I'd say that Long Beach - Hong Kong will probably work out cheapest, all included.
Good luck, let us know how you get on and anything else, just ask.
