Mix graham cracker crumbs, melted butter, and sugar in a bowl.
Press the mixture into a 9-inch springform pan and bake at 325°F (163°C) for 10 minutes.
Beat softened cream cheese and sugar together until smooth.
Add eggs one at a time, followed by sour cream, vanilla extract, and lemon zest.
Gently fold in fresh blackberries into the cheesecake batter.
Pour the filling over the crust and place the pan in a water bath.
Bake for 60-70 minutes, then let the cheesecake cool in the oven for 1 hour.
Prepare the topping by simmering blackberries, sugar, and lemon juice for 10 minutes.
Refrigerate the cheesecake for at least 4 hours, then serve with the blackberry topping.