Preheat oven to 325° F. Line an 8x8 baking dish with nonstick foil or parchment paper.
In a medium-sized bowl, mix graham cracker crumbs, sugar, and melted butter until combined well. Pour into prepared baking dish and press down evenly.
Place in oven and bake for 10 minutes. Remove and set aside.
Make puree by adding blackberries and sugar to a bowl then use an immersion blender to puree. Or add to a food processor and blend until smooth.
Place a fine mesh strainer over a bowl then pour blackberry puree into it. Let strain completely pushing with the back of a spoon to help liquid strain through.
In a large bowl (or in a stand mixer), add cream cheese, sugar, eggs, sour cream, flour, vanilla, lime zest and juice. Use a hand mixer to combine until smooth and creamy. Pour over graham cracker crust.
Drizzle strained blackberry puree in lines over the cheesecake filling. Use a toothpick to swirl the puree. Place in oven and bake for 30 minutes or until filling is mostly set. Should still have a slight jiggle to it.
Let cool at room temperature for 20-30 minutes then place in the fridge to chill completely before serving.