Preheat the oven to 325°F (160°C). Grease the bottom of a 9-inch springform pan.
In a mixing bowl, combine the graham cracker crumbs, sugar, and melted butter. Mix until the crumbs are evenly coated. Press the mixture firmly into the bottom of the springform pan to form the crust. Bake for 10 minutes, then let it cool.
In another bowl, beat the softened cream cheese with an electric mixer until smooth. Gradually add sugar and continue to beat until well combined. Mix in the vanilla extract.
Add the eggs one at a time, mixing on low speed after each addition until just combined. Gently fold in sour cream with a rubber spatula until smooth.
Pour the cheesecake filling over the cooled crust in the springform pan. Smooth the top with a spatula.
Bake in the preheated oven for 60 minutes, or until the center is set and slightly jiggly. Turn off the oven, crack the door, and let the cheesecake cool in the oven for about 1 hour.
While the cheesecake is cooling, prepare the blackberry topping. In a saucepan, combine blackberries, sugar, lemon juice, cornstarch, and water. Cook over medium heat, stirring gently until the mixture starts to bubble and thicken, about 5-7 minutes.
Remove the blackberry mixture from heat and let it cool slightly. Once cooled, pour it over the cheesecake, spreading it evenly.
Refrigerate the cheesecake for at least 4 hours, or overnight for best results. Once set, carefully remove the sides of the springform pan.
Slice and serve chilled. Enjoy your delicious Blackberry Bliss Cheesecake!