How can I clean cloth lampshades?

I simply use a damp cloth and wipe the dust off. If I find a stain it's a tide pen and damp cloth.

I wash my cloth lampshades in the kitchen sink and hang them on the clothes line to dry.

If it is badly in need of a wash and a pen or lint roller or damp cloth won't solve it, you could try getting a clean container with an opening quite a bit bigger than the shade, put some mild soapy warm water inside the container and dip the shade in and leave a minute and out and in again a few times and then another container to rinse. This works for a silk lampshade, according to what I read on line when I was planning to clean my old lampshades. I never got around to trying it. I think it will work with your lamp.