One big problem for new-born kittens, especially at this time of year, is fleas. A bad enough infestation can literally suck the kitten dry
Frontline spray can be used on kittens from two days old. Just spray some on some cotton wool and rub them all over with it. You'll need to see a vet to get it though, you can't just buy it. And it is fairly expensive, around £15 quid for the bottle, but one treatment lasts for around 3 moths, so one bottle goes a long way.