To make the simple syrup, add one cup of the water and the cup of sugar to a large saucepan. Heat and bring to a boil, stirring occasionally, until the sugar has fully dissolved. Remove from the heat and set aside.

Add the blackberries to a blender (or food processor) and blend until smooth. Strain through a fine-mesh strainer and discard the seeds.

Add the simple syrup, blackberry juice, remaining 4 cups of cold water, and fresh lemon juice to a large pitcher. Mix well to combine.

Refrigerate until ready to serve.