Watch the Some Notes About Chet online guitar lesson by Muriel Anderson from Essentials: Chet Atkins Style
As you know, Chet played most of his music on the electric guitar, and that's how he developed his signature sound using a thumbpick with a muffled bass. He later played a lot on a nylon stringed guitar, which was easier on his fingernails. You can either play with your fingernail or a thumbpick, either is great, or if you prefer just to do it with the side of your thumb, that will work as well. Any type of instrument you have will work; we can all play music!