I've found some of my apartments on booking.com and some on VRBO.com - never had any problems with either one. Booking.com is the easiest one as you get an immediate confirmation and usually don't have to pay until you stay, although that varies with the particular accommodations. I did a search using sample dates in July and came up with 38 apartments that sleep 2 adults and 2 children (don't know the ages of your kids, they may be considered adults for these purposes).
We'd need more information regarding ages of kids and budget to give you better suggestions.
Hopefully someone will give you some good information about which neighborhood is more convenient to transportation from Tegel. What I found when I stayed in Berlin is that public transportation is excellent and you can usually get from anywhere to anywhere else without too much inconvenience so that may not even be an issue.