There is a genetic component to bipolar, so your child is more likely than the general population to get it, but it is by no means guaranteed. If both sides of the family are full of severe mental illness, then it's even more likely, but the incidence of bipolar in the population is low enough that even a significant increase in chance is still a pretty low chance.
It could be. I have a friend who's grandma and mom are bipolar and he is too.