Absolutely discourage this for sure or he may wind up a big adult that is bitey! You don't want him to ever think of hands and fingers as toys.
You can do several things:
1. Give him a toy made for biting and kicking when he starts doing this to redirect his attention and give something more appropriate to use.
2. Remove yourself for a couple of minutes every time he does this. Eventually he will learn biting = no attention, which is what he's looking for.
3. You can also try to gently put him down and say "OUCH" if you are holding him or playing on an elevated surface, and again ignore for a minute or so.
4. I have found that lightly tapping on their forehead and saying "NO" in a stern way will get their attention too. And again, stop the interaction for a minute or two.
5. In general, do not play games with your hands. Wand type toys are best for both exercise and mental health. They should be focusing on objects, not people, to get their crazies out LOL
Kittens should not ever be punished for any behavior which is in their nature but rather redirected and shown what are the appropriate ways to do these behaviors.
Eventually they do calm down and if you are consistent in applying methods to stop and prevent biting/kicking they will begin to learn not to do it with humans. It will take time, don't expect overnight though. they are after all, babies still.
I definitely don't want him to do that as an adult. My boy, Lucky is like that so I figure that was because that behavior was not discouraged when he was a kitten.
Thank you very much for these instructions!
Yes, even the average adult cat doesn't understand punishment.