With a vegetable peeler remove zest from 4 lemons and squeeze enough juice from these and remaining 2 lemons to measure 1 cup.
In a saucepan boil 2 cups water with sugar, stirring, until sugar is dissolved. Add zest, lemon juice, and remaining 2 cups water and cool.
In a food processor or blender puree blackberries and stir into lemonade. Pour blackberry lemonade through a sieve into a pitcher or other container and chill. Chill lemonade, covered, at least until cold and up to 2 days.
Serve lemonade over ice in tall glasses, garnished with lemon slices.