I have 7 dogs, had 4 cats, and am taking care of a bunch of strays. Bathing in dishwashing liquid will kill adult fleas and is safe for the cat. Do not put several different things on your pets. It will make it sick. Stick to one treatment for the cat and one for the environment. I use pills from the vet for the cats. You give the animal one pill a week. Then go to any department store like Walmart and get Sergeants carpet powder that kills flea eggs, larvae, and adults. Lightly dust the cats bedding, around the baseboards, and furniture. Wait a couple of hours and vacuum. Repeat at weekly intervals. If you can wash the pets bed, hot water will kill the fleas. Cats are more sensitive to chemicals so be careful. If you see the cat drooling or foaming at the mouth, wash with soap immediately and give a 25 mg tablet of benadryl. Take to a vet. For outside, if you have a yard, get a good yard treatment and use it as directed. Their is a spray that kills eggs, larvae, and adult too. Only use the powder or the spray. Not both.
You can also set out a pan of water with a squirt of soap under a night light or other light source. Turn off all other lights at night and the fleas will jump at the light, hit the water, and die. Works surprisingly well to get rid of the adult fleas.