Dawn dish soap is a very good cleaning soap and works to kill fleas as well as removing grease and other dirt.
If you don't have any carpet cleaning agents to clean your carpet then yes you can just use some Dawn Dish Soap or other dish soap in your Carpet Cleaner and it will do a great job cleaning your carpet.
Another good thing of Dawn Dish Soap like I mentioned above is that if you have pets in your house and they get fleas on the carpet or upholstery you can use the Dawn Dish Soap in the carpet cleaner to kill fleas.
Dawn Dish Soap will also kill fleas on your dog or cat and is a safe soap to wash your dog or cat with to get rid of fleas and other dirt and grime from your pet.