Pour the sugar and water into a medium saucepan. Heat over medium to medium-high heat, stirring occasionally, until the sugar is just dissolved.
Remove from heat and set aside to cool while you make the puree.
Place the blackberries and sugar in a food processor. Puree until the fruit is completely broken down.
Transfer the puree to a fine mesh sieve placed over a bowl. Use a rubber spatula to press the puree through the sieve so the juice goes into the bowl and all the fruit pulp and seeds are left in the sieve. There should be almost 3/4 cup of blackberry juice. Discard the pulp and seeds
To a 2.5 quart pitcher, add the lemon juice, water, simple syrup, and blackberry puree. Stir together until evenly combined.
Serve over ice or refrigerate until ready to serve.
If you like your lemonade more sweet than tart, increase the amount of sugar to 1 cup.For days when you're short on time, you can skip making the simple syrup. Pour the lemon juice, water, sugar, and blackberry puree into the pitcher and stir really well. Sugar doesn't like to dissolve in cold water.