Thingstodo, combinations can be perfect or horrible depending on the carpet and problem. Tough to know without having the problem in front of us.
Dish soap is pretty harsh. It'll remove everything, including any guard product and even some colorings. It'll certainly take out grease but it's a shotgun soap. Good for stripping wax from the car before a detailing. Elphaba can probably tell us more about side effects on different carpet materials.
Also, I wouldn't combine peroxide unless I had a large broadcast area that needed the same treatment. Better to spot with each as needed. Remove what you can of the grease - emulsify & extract - then if there's a stain go ahead and tune a separate peroxide mix to the stain and carpet color. I'd use something else though, like Advantage or similar.
This is from my limited experience cleaning family residential rental carpets. Zero training. Ephaba can probably correct everything I'm saying too.