Combine defrosted or fresh blackberries with 1/3 cup sugar and lemon juice in a saucepan. Using a potato masher or whisk break up blackberries and bring to a full boil, turn to simmer for about 7-9 minutes or until slightly thickened with a similar consistency as corn syrup. Set aside.
To make crust: Crush graham crackers with a rolling pin to obtain 1 1/4 cup. Add melted butter and combine well. Press onto the bottom of a 9-inch diameter springform pan and place in oven for 10 minutes. Remove from oven and let cool slightly. Reduce oven temperature to 325°F.
Once cool enough to handle, place springform pan on 2 layers of aluminum foil and wrap outside of pan tightly with foil. Ensure that no water could penetrate thru foil. Place in a roasting pan and set aside.
Blend cream cheese and sugar on lowest speed, alternatively mix by using a hand held spatula until no lumps remain. Scrape bowl well. Add lemon zest and one egg at a time waiting for previous added egg to be fully incorporated and mixture shows no lumps. Scrape bowl and beater between additions.
Add sour cream and vanilla. Scrape about 1/3 of cheesecake batter into prepared pan. Swirl some blackberry filling into it. Add another 1/3 of the cheesecake batter followed by more blackberry filling. Spread remaining cheesecake batter on top. Place into roasting pan.
Pour water into the roasting pan about 1-inch high and place in oven.
Bake until center of cake no longer trembles when tapping sides of the pan, about 70-85 minutes but it could take slightly longer depending on temperature of cheesecake batter.
Any remaining blackberry fruit spread can be used as a sauce when serving the cheesecake. If desired serve with crème Chantilly.
