My laptop has two stains on it from me resting my palms on it while typing. I’m thinking maybe hydrogen peroxide might take the stains off the plastic. Any ideas?
Mr. Clean Magic Eraser
Even if it doesn’t work, you can use it on 200 other spots and stains around the house 
Don’t use an organic solvent like alcohol or such. Plaskic and finishes may be ruined. I use a soft bristle paint brush dampened with water and pressed dry in a towel to clean the keys and cracks on equipment. The brush is slightly damp and I dab the brush down between the keys to be cleaned. Go side ways on a crack with the brush. This won’t work if your problem is it’s a stain and not just hard to reach the dirt.
My laptop has a permanent mark where my left palm rests - I suspect the plastic is permanently deformed from the heat of the hard drive below; some stains just aren’t cleanable, unfortunately.
Try unscented baby wipes (i.e., Huggies.) I’m serious; I use them for household cleaning almost everywhere. The mild detergent cleans easily without attacking plastic. The soft cloth won’t scratch and keeps the cleaner from dripping into the case…