June is the peak of Summer up there I think - it differs in different parts of India. The monsoon will have hit by August which keeps things cool, but it can be humid. August should be a better time than June in terms of heat though - it isn't like Europe. The main problem in the monsoon is that is when many of the major health issues develop. You need to be very cautious with food and water, perhaps also look into typhoid vaccinations for the kids (Hep A if the haven't had it) (at least talk to a doctor about it). There will also, most certainly, be alot of dengue about, although some of the northern regions may be OK. Last year there were also malaria outbreaks in several places where you don't usually here much about it. Every year there is cholera around, typhoid, hepatitis outbreaks.
I would ask on the India branch about travelling in those regions if it is still monsoon. It could be tough if the infrastructure isn't good, but perhaps that adds to the adventure? If it could be a big problem then perhaps the better developed places may be better for getting around. I know someone whose train was delayed 36 h during the monsoon once. That wouldn't be much fun with young kids (although the trains themselves are great with kids).
I haven't seen the north yet so can't give any specifics sorry. We visited Rajasthan with the kids, and several places in the south (where we live).