How do I clean my Wegner CH30 chairs?  

see WATER stains thread
Try reading the water stains thread

Depends on what you're cleaning off
I've cleaned a ton of these chairs - they're easy to do! If it's just age and grease and dirt, there's a product called "Murphy's Oil Soap" which is great for removing greasy gunk from these chairs. You can get it from any hardware store, or even the grocery store (in the USA).
Get a bottle (water it down if it's the concentrated version) and remove the seats so you're only dealing with the wooden frames. Apply the solution with a sponge or spray bottle, and let it sit for 10-15 minutes. Then agitate with a plastic-type scrub pad and rinse with clean water. Repeat if necessary. Pat dry immediately with a towel and let dry *thoroughly*. Once it's dry, you can go over it with fine sandpaper to smooth out the grain some (anything over 200-grit is safe) then brush off the sanding dust.
The teak should look grayish, and will take teak oil nicely. In fact, that's one of the great joys of doing this, as you can see the teak drink up the oil and regain its beautiful glow! If you want a really smooth surface, work the oil in with fine steel wool or a CLEAN scrubbie pad. Then re-oil it over the course of a few weeks, whenever it starts to look dry, and it will rebuild a nice patina. Be sure to give it a quick dusting before you oil it, or you'll just drive the dust into the grain again and it will start to look dirty.
If there's any kind of varnish or other permanent finish, you'll need to strip it off with stripper. I prefer a citrus-based stripper as it's non-toxic and won't pollute. If the grain is super-grimy, you can use a fine brass-bristled brush to work the stripper into the grain. Teak is a hardwood, so it can take a lot of this and still look great. Refrain from any desire to seal it with varnish, as it'll only make it harder to clean the next time - teak looks better with an oil finish anyway.
If there are any water stains or other blotchy areas remaining after the initial cleaning, they can be improved with a wood bleach (oxalic acid) solution in hot water. Be sure to remove ALL the crystals that result before you oil it!
It's a nice sunny-afternoon project, and you'll love the way they look when they're done. Have fun!

Great guide 🙂 sounds like a ...
Great guide 🙂 sounds like a very rewarding process

I use Murphy's too, but..
..I use it straight out of the bottle, applied with a folded paper towel. I leave it on for only a minute or two, then wipe off with clean paper towels. Most grime will come off in this first minute. Then I rinse off the soap residue with a wet sponge very quickly, then dry.
I just got a single CH23 chair last night at auction for ten bucks--woohoo! I cleaned off half the back and will admire the difference a bit before doing the rest of it. I like before 'n' afters. They can be so dramatic.