Preheat oven to 350°F. In a large bowl, beat cream cheese and 3/4 cup sugar with an electric mixer on medium until smooth. Add eggs, lemon juice and 1 tsp. vanilla; beat until blended. Pour into crust. Bake until edges are set and center jiggles a bit, about 30 minutes.
In a medium bowl, stir sour cream, 1/4 cup sugar and 1 tsp. vanilla. Remove cheesecake from oven; let cool 5 minutes. Spread sour cream mixture on top and return to oven. Bake until topping is firm, about 15 minutes. Let cool 20 minutes on a wire rack, then cover and chill 4 hours or overnight.
If desired, place half of blackberries in a medium bowl, sprinkle with 1 tsp. sugar and lightly mash. Let stand 10 minutes. Arrange all berries on cheesecake.