Preheat your oven to 350°F (175°C). Line a muffin tin with cupcake liners.
In a mixing bowl, whisk together the flour, sugar, baking powder, baking soda, and salt.
Add the softened butter and beat until the mixture is crumbly.
In another bowl, whisk together the eggs, lemon zest, lemon juice, and buttermilk.
Gradually add the wet ingredients to the dry ingredients, mixing until just combined.
Divide the batter evenly among the cupcake liners, filling each about ⅔ full.
Bake for 18-20 minutes, or until a toothpick inserted into the center comes out clean. Let cool completely.
For the frosting, beat the cream cheese and butter together until smooth.
Gradually add the powdered sugar, beating until fluffy.
Mix in the blackberry puree until well combined.
Frost the cooled cupcakes with the blackberry cream cheese frosting.