Preheat the oven to 325°F (160°C).
In a mixing bowl, combine graham cracker crumbs, melted butter, and sugar until well-mixed. Press the mixture into the bottom of a 9-inch springform pan to form the crust. Bake for 10 minutes and let it cool.
In another bowl, beat the softened cream cheese and sugar using an electric mixer until creamy and smooth. Add vanilla extract and mix well.
Add the eggs one at a time, mixing well after each addition. Then, mix in the sour cream until incorporated.
Pour the cheesecake filling into the cooled crust and smooth the top. Bake for 40 minutes or until the center is set but slightly jiggles. Turn off the oven and crack open the door, allowing the cheesecake to cool for 30 minutes in the oven.
Remove the cheesecake from the oven and let it cool to room temperature. Then, refrigerate for at least 4 hours, preferably overnight.
To prepare the blackberry topping, combine blackberries, sugar, lemon juice, cornstarch, and water in a saucepan. Cook over medium heat, stirring gently until the mixture thickens, about 5-7 minutes. Let cool.
Once the cheesecake has fully chilled, spread the blackberry topping over the surface. Slice and serve.