Preheat your oven to 350°F (175°C).
In a bowl, mix the graham cracker crumbs, 1/4 cup sugar, and the melted butter until well combined. Press this mixture firmly onto the bottom and up the sides of a 9-inch springform pan. Bake for 10 minutes, then allow to cool completely.
In a saucepan over medium heat, combine the blackberries, 1/4 cup sugar, and lemon juice. Cook the mixture until the berries soften and the sugar dissolves, about 10 minutes. Once cooled, press it through a fine sieve to remove the seeds, collecting the coulis in a bowl.
In another bowl, beat the cream cheese and 1 cup sugar until smooth. Add the eggs one at a time, mixing until just blended each time. Mix in the blackberry puree and vanilla extract until the color is uniform, with no streaks remaining.
Pour a thin layer of blackberry coulis over the cooled crust. Let it set in the fridge for 10-15 minutes. Then, carefully pour the cheesecake filling over this layer and bake at 325°F (160°C) for about 50-60 minutes, or until the center is set but still slightly wobbly.
Allow the cheesecake to cool completely. Refrigerate for at least 4 hours, or overnight for the best flavor and texture.
Once chilled, spread a thin layer of blackberry coulis over the top and decorate the edges with fresh blackberries.
To serve, remove the cheesecake from the springform pan and slice it with a hot knife for clean cuts