How should I clean a white Eames shell chair?  

cleaning products
Yeah, the fiberglass is pretty much impervious to everything except VOCs. Bleach won't hurt it a bit. The best thing I've found for cleaning them is a product called "awesome," that's available at dollar stores. I don't know what's in it, but it cuts through all kinds of grime, and it's cheap.
The one stain that nothing (as far as I know) will get rid of is shock mount bleed. That's the dark spots that appear on the seat, directly over the shock mounts. It's most often found on early, light colored chairs. Anyone have a secret for getting rid of those?

You're not supposed to
You're not supposed to get rid of shockmount bleed. It was Charles' opinion that it represented the "honesty" of the material. And, since it is basically epoxy that has seeped up through the material my guess would be nothing can get it out completely.
Also, in cleaning the shell - avoid anything with abrasives as those will cut through the gelcoat. LRF has good results with plain old pain thinner.
I had a white shell and two parchment ones. Nothing was able to get all the dirt up. After a while I began to think of it as part of the charm of their age.

Thanks, Poach!
I found Totally Awesome at Dollar Tree. I hate to sound like a late-night pitchman for the product, but it did a great job removing some of the chair's yellowing as you can see in this pic I took halfway through cleaning it.

Hail, Lucifer!
You make a good point about letting the superficial dark lines speak to its history.