For the Blackberry Puree:

1. Combine blackberries, sugar, and lemon juice in a saucepan over medium heat. Cook until the berries break down and release their juices, about 10-15 minutes.
2. Strain the mixture through a fine mesh sieve to remove the seeds, pressing to extract as much liquid as possible. Set aside to cool. Divide the puree into two portions: one for the filling and one for the layers/topping.

For the Crust:

1. Preheat your oven to 350°F (175°C). Mix graham cracker crumbs, sugar, and melted butter until well combined.
2. Press the mixture into the bottom of a 9-inch springform pan, forming a compact, even layer.
3. Bake for 10 minutes, then remove from the oven and let cool. Reduce the oven temperature to 325°F (163°C).

First Layer of Blackberry Puree:

1. Once the crust has cooled, spread a thin layer of blackberry puree over the crust. Place in the freezer to set slightly while you prepare the filling.

For the Cheesecake Filling:

1. Beat the cream cheese and sugar together in a large bowl until smooth. Mix in the sour cream and vanilla extract.
2. Add the eggs one at a time, beating just until blended. Do not overmix.
3. Fold in 1/2 cup of the blackberry puree to achieve a light purple color.
4. Pour the filling over the set blackberry layer on the crust, smoothing the top with a spatula.

Baking:

1. Place the springform pan in a large roasting pan. Fill the roasting pan with hot water halfway up the sides of the springform pan to create a water bath.
2. Bake at 325°F (163°C) for about 60-70 minutes, or until the edges are set but the center is still slightly jiggly.
3. Turn off the oven, open the door slightly, and let the cheesecake cool in the oven for 1 hour to prevent cracking.
4. Remove from the oven and water bath, then chill in the refrigerator for at least 4 hours or overnight.

Final Layer of Blackberry Puree:

1. Once the cheesecake is fully chilled and set, spread the remaining blackberry puree over the top. Return to the refrigerator to set before serving.

To Serve:

1. Run a knife around the edges of the cheesecake to loosen it from the pan before removing the sides.
2. Slice and serve chilled, garnished with fresh blackberries if desired.