Amsterdam is child-friendly but definitely not car-friendly. Parking is super expensive (5 euros per hour in the center), hard to find parking spots, and traffic is utter chaos with trams, confusing lanes that suddenly diverge or converge, lots of one-way streets, cyclists ignoring all traffic rules and coming from all sides; but in case you hit a cyclist (even if the cyclist ignored a red light), insurance-wise you are the one who's going to pay. Even Dutch people from outside the city are daunted by the traffic here. My advise is to avoid driving in Amsterdam, especially with a British car (because you may have bigger dead angles and bigger chance of overlooking cyclists and mopeds).
The best option is to leave the car at one of the special Park + Ride parking lots at the edges of town (8 euros per 24 hours) and use public transport. iamsterdam.com/en-GB/…parking
Or stay at a hotel outside the center, for example the district called Noord (North) may have hotels with free parking, and I believe the NH hotel there has a free shuttle into the center.
What is the purpose of your trip? Obviously there is more to the Netherlands than just Amsterdam so I am wondering why you want to go to Amsterdam. Don't get me wrong, I love my city and I think there is loads to see and do here for kids too. But you may prefer staying in a smaller town with free parking.
And how old are your children?
For now, this page may help find fun things to do with kids: iamsterdam.com/en-GB/…families-and-children
Edited: 9:27 am, September 20, 2013