Kittens love to play, and they will play all the time until they run out of energy, then they sleep. But when they wake up, it's time to play again. When kittens play, they play aggressively, and part of that includes biting and scratching. It is natural behavior for them. They are preparing to become hunters, so they pounce on anything that moves with their claws, and bite it. If the mother cat is present, she will teach the kitten with a growl or swat when the play becomes too aggressive. If not, then the human owner has to perform this function.
Kittens are so cute that humans love to play with them. Since their claws and teeth are so sharp, it's inevitable that you are going to get some scratches during play. That's one of the ways you can identify kitten owners, by all the scratches on their hands. The kitten will calm down and learn the limits of play eventually, but it seems like it can take a long time. Here are some ways to stop kittens from biting.
First of all, it will take the kitten some time to learn when not to bite and scratch. I recommend keeping a pair of leather gloves handy when you want to play with the kitten to protect yourself from the biting and scratching. Anitra Frazier in her book The Natural Cat recommends never playing with a cat with your hands, period. She says to always use a toy, otherwise anyone that reaches out to the cat with a bare hand is liable to get scratched.
Next, you have to take the mother cat's place and establish yourself as the one who sets limits. If the kitten grabs or bites your hand during play, say "No" sharply and give a tap on the nose to get him to back off. This doesn't work with all kittens, but always give the sharp "No" and use another method if the nose tap is interpreted as play. The key here is to issue a sharp and firm sound when unwanted behavior starts. Before long the kitten will understand it. You want them to stop the bad behavior, but not to run off across the house. If you can get them to stop but continue to play within your limits, the training goes much faster because you are able to reinforce good and bad behavior.
Toys are Helpful
A sharp "Ssst" in the kitten's face is a little closer to what he would get from the mother cat if his behavior gets too aggressive. This can replace "No". Sometimes kittens respond better to Sssst, just do it sharply.
Cover the ears
If you don't want to tap the kitten on the nose because you'd rather not, or because they think you are playing, try putting the palm of your hand over the ears and holding them down. Cats and kittens hate this because it takes away one of their most important senses and means of detecting threats. This is a good way to stop bad behavior patterns like biting.
I have trained 5 different cats using a hand clap to indicate unwelcome behavior. The sharp noise gets their attention, and when combined with "No", they get the message before long. After a while, the hand clap alone makes them scatter. This can work with kittens that like to ambush your bare feet when you walk by.
Substitute a Toy
To let kittens pounce and play, get a few cat toys. A feather or mouse on the end of a string as pictured here lets them get some aggression out while protecting your hands from a distance. We always have a few furry fake mice on the floor for them to bat around and stalk and chase. A scratching post or cardboard scratching box is helpful too. This gives them a place to dig their claws in and stretch.
When a kitten is tired is a good time to reinforce behavior limits. It is hard to pet them when they are active because they want to scratch and play. Pet them when they are tired and they will get used to you touching them. This way they won't attack you every time you try to do it.