Dependency property updates are handled via property metadata, which is defined as part of your DependencyProperty. (It can also be added to existing DPs, but that's another topic.)
Define your DependencyProperty with metadata:
public static readonly DependencyProperty MyValueProperty =
DependencyProperty.Register("MyValue", typeof(object), typeof(MyControl),
new UIPropertyMetadata(null, new PropertyChangedCallback(MyValue_PropertyChanged)));
Then implement your callback:
private static void MyValue_PropertyChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
{
MyControl c = (MyControl)d;
c.DoSomething();
}