There is no one cheapest airline or flight. You need to check comparison search engines such as Kayak, Vayama, and SideStep as well as brokers like Expedia and Orbitz and compare prices. Also, sign up for Travelzoo alerts and you'll get notifications of the best deals. It's usually cheapest to buy about 3-4 months out. Last minute deals are not really "deals" - they're only available when an airline realizes last minute that they have a lot of unsold seats for a particular flight, so you can't rely on that happening. China in the summer is very hot and it's technically lower season. So you should be able to get deals.

Also check out student travel sites. Some of their fares do not require that you are a student and can be a lot cheaper than you'd find elsewhere. (There is a Travel Cuts in Portland and there is also STA online)

I'd agree with D.G. Look at the travel sites. The bigger the airport you're willing to fly from (i.e. SeaTac vs PDX... or even SFX) the lower prices tend to get. Keep an eye out for Asiana Airline, they're Korean and offer daily flights to china, usually through Seoul. Affordable, and very decent quality.

What is affordable? Just head to kayak.com or Expedia.com and get alerts for when prices on that route drop. I imagine if you can fidn a cheap RT to LA you might even get cheaper overall prices for a RT to LA and go to Shanghai from there? Since you need to leave soon, I would say also go visit a travel agent somewhere like Flight Centre as they will know the best options.