Here is a video demo for auto detailing. Shampooing car carpets can be difficult to do, but this DIY expert presenter breaks it down into simple steps. He did remove the car seats prior to cleaning the car carpets, but it’s optional.
The presenter used very simple equipment to make it work. This tools he used included a bottle of foaming carpet cleaner, powered scrub brush and wet/dry shop vac. The powered scrub brushhead itself is less than $20, but it needs to be attached to a drill (another $50). This is optional if you are willing to scrub the carpets manually.
This approach is interesting, as it emulates what a portable carpet cleaning machine does. If you do not already have a wet/dry shop vac and a powered drill or carpet/upholstery brush, getting a car carpet cleaning machine may just be easier. These portable machines combines brushing and suctioning into one swipe. Some like the Rug Doctor Portable comes included with a motorized brush-head. These machines are easy to use and can also clean car upholstery.
You may want to check out this other video from someone who used to professionally do auto carpet cleaning. He shows home users how to do it on their with a low cost extractor. Auto Detailing with the Bissell Little Green.