Preheat the oven to 325°F (160°C).
In a medium bowl, combine graham cracker crumbs, melted butter, and sugar. Mix until fully combined.
Press the crumb mixture firmly into the bottom of a 9-inch springform pan to create an even layer. Bake for 10 minutes, then remove from the oven to cool.
In a large mixing bowl, beat the softened cream cheese and sugar with an electric mixer until smooth and creamy.
Add eggs, one at a time, mixing well after each addition. Stir in vanilla extract, sour cream, and blackberry puree until combined.
Pour the cheesecake filling over the cooled crust in the springform pan, spreading it evenly.
Bake for 50-60 minutes, or until the center is set but slightly jiggly.
Turn off the oven and crack the oven door, allowing the cheesecake to cool for about 1 hour before transferring to the refrigerator.
Chill the cheesecake in the refrigerator for at least 4 hours, preferably overnight.
To make the topping, combine fresh blackberries, sugar, and lemon juice in a saucepan over medium heat. Cook until the berries soften and break down, about 5 minutes. Remove from heat and let cool.
Once the cheesecake is chilled, pour the blackberry topping evenly over the cheesecake before serving.