Best of luck finding a spot for a child under six. Berlin is great in that the government offers "free" kita (kindy) places to kids between 3-6 years, but finding a place is ridiculously hard. My son was three when we moved to Berlin and it took about six months before he was able to start--and that was in an all German kita. Some children apparently do quite well in immersion environments, but for us it was a disaster.
Dahlem is lovely, though a little bit far out. The Berlin Metropolitan School is income based, but cheaper than a number of the international schools. It's near Friedrichstrasse and seems fairly decent. There weren't any places for my son, but he's now a bit older than your son and perhaps there's a place in their kindy group for him. These types of schools often keep places open for people who have newly moved to Berlin, by the way, so it's easier to get in when you've just arrive than when you've been here a while. :) Their kindergarten is bilingual, with two teachers in each class: one has to be a German native speaker and one has to be an English native speaker. They seem quite nice.
If you're getting a car, the Berlin British School has a very popular kita. It's quite a trek out from Dahlem if you're catching the train, but they have a good reputation. Warning: the Berlin British School has fees significantly higher than a lot of other kindergartens.
Somebody recommended the Schele Schule to me. They're also private and bilingual. I didn't really look into them, but it might be an option.
Aufgepasst (www.aufgepasst.de/en) is another bilingual kindergarten. They didn't have a place for us so I've never been there, but a lot of people recommended it to me. I don't believe Aufgepasst is private as such.
Sara Kita (www.sara-kita-berlin.de/) opened fairly recently and have been advertising that they actually have places! Another bilingual kita with two teachersone English and one Germanin each class, it's again a little far from you but if you have a car it might be an option. This one isn't exactly private, but there's a small fee every month on top of the kita-gutschein (what the government pays).
I'm sure there are others, but those are the ones I can think of off the top of my head. Best of luck!