Your kitten is behaving in a normal kitten manner. He is playing now, and you should never allow him to bite your fingers, or use your hand as a play toy. Kittens need to learn that biting at any time is not acceptable behavior.
Redirect the play. Get some cat toys that are soft and appealing to your kitty. When he starts biting, put him on the floor and toss the toy. Praise him when he is playing with it. Never snap your finger at your kitty's nose or face when teaching him not to bite. A firm "NO" and redirect is the best way to go.Also never hit your kitten under any circumstance. That will only get him scared of you.
Also, your kitten will be going through teething phases, so that is another time he may be more attracted to chewing on your fingers. Here is some more information about teething in kittens and time periods for this stage of develooment.
Buy him bigger cat toys or even small dog toys. The last cat I had would play aggressively like that so I bought her a bunch of small dog or puppy toys to play with. She loved them. She would pounce on them, bite and kick them while holding them with her front paws. Sounds like your cat is a little bored and wanting to play so he plays with you since he doesn't have anything else to play with.
They Don't even have to be expensive.
You can use an empty Toilet Paper Roll but that doesn't help to curb their natural Tendency to go for the Full Roll!
Ball of Yarn, wrap around the outside of a Couple Fingers, 20 times or so. Tie a Piece tightly around/across the Outside of the Center of the Roll. Cut straight through the Center of the ends. Fluff up.You've just made a Pompom ball!
Finger Knight a Strip about 4' long or so. Finish off & Toss to Kitty.
Egg Carton Cups, cut apart. You can even add Pipe cleaners stuck through to make their Own personal Spiders.
That "Security" tab/ring around Plastic Milk Bottles. Poof Open your Beverage/Milk & have a Free/instant Toy!