In a small saucepan, combine the blackberries, sugar, and 1/2 cup water. Place over medium heat and simmer for about 8 to 10 minutes. As the berries soften, mash them gently with a spoon to release their juices and create a thick, purple syrup. This step builds the base flavor of your lemonade.
Remove the saucepan from the heat and carefully pour the mixture through a fine-mesh sieve into a bowl. Use the back of a spoon to press the pulp and extract as much liquid as possible. Discard the solids and allow the syrup to cool completely before moving to the next step.
In a large pitcher, combine the cooled blackberry syrup with fresh lemon juice. Stir well until the mixture is evenly blended. Slowly add the 4 cups of cold water, stirring as you pour. Taste the lemonade and adjust with more syrup or lemon juice if you prefer it sweeter or tarter.
Fill glasses with ice cubes and pour the blackberry lemonade over them. Garnish each glass with a lemon slice and a few fresh blackberries for extra flavor and presentation. Serve immediately while cold, or refrigerate the pitcher for up to 2 days before serving.