Preheat your oven to 325°F (163°C). Grease the sides of a 9-inch springform pan with butter or non-stick spray.

In a mixing bowl, combine graham cracker crumbs, sugar, and melted butter. Stir until well mixed. Press the mixture firmly into the bottom of the prepared pan.

Bake the crust in the preheated oven for 10 minutes, then remove and let cool.

In a large mixing bowl, beat the softened cream cheese with an electric mixer until smooth. Gradually add in the sugar, mixing until well combined.

Add eggs one at a time, mixing well after each addition. Then add the vanilla extract and lemon juice, and mix until just combined.

Pour the cheesecake filling over the cooled crust in the springform pan. Smooth the top with a spatula.

Bake the cheesecake in the oven for 50-60 minutes, or until the center is set but slightly jiggly.

Allow the cheesecake to cool at room temperature for 1 hour, then refrigerate for at least 4 hours or overnight.

In a saucepan over medium heat, combine blackberries, sugar, cornstarch, lemon juice, and water. Stir gently and cook until the mixture begins to thicken and bubbles, about 5-7 minutes.

Once thickened, remove the topping from heat and let cool. Spread the blackberry topping over the chilled cheesecake before serving.

Carefully remove the sides of the springform pan and slice to serve.