The problem is light is being reflectecd form the screen onto your subject. That's a basic lighting issue for keying. Not much you can do about it After Effects. Spill suppression is a function of better keyers such as Keylight and, if you visit the Keylight site or search for tutorials for the product, you will find lots of help on using the filter. But properly lighting a key set is fundamental to getting a good key. And that, too, is easily researched. You need room between the subject and the green screen and you need a light meter and you need control media for your lighting kit in the form of neutral density filters or dimmers (which cause the temperature of the light to change, a different issue).
Keylight has a great spill suppression system.
Rats, I should have known that, for some, a complex issue like this requires only a one-click solution.
There you go, all you need is software. You don't need a competent lighting setup or experience.