Pour through a fine mesh sieve using a silicone spatula to work it through until you have a 1/2 cup. Discard seeds and pulp.
In a glass bowl, whisk together sugar, lemon juice, eggs and blackberry puree (everything but the butter).
Place over a pot of simmering water and cook, whisking occasionally, until thickened and able to coat the back of a wooden spoon (about 10 minutes).
Remove from heat and allow to cool for about 10 minutes.
Cube the butter into 1/2" pieces and stir into the curd until completely combined.
Place a piece of plastic wrap against the surface of the curd and refrigerate until set.
Notes
Under-ripened blackberries: If your fresh blackberries aren't at peak ripeness, you can help them release their juices (so you have enough for your curd) by cooking them over medium heat for a few minutes.
Use a non-white spatula: Berry juice has a tendency to stain, so I recommend using a silicone spatula that isn't white when working the puree through the sieve.
Fresh lemon juice: Freshly squeezed juice works best for this recipe.
Testing for doneness: The curd has cooked long enough when it coats the back of a wooden spoon.
Storage: Blackberry curd can be refrigerated for up to 7 days and frozen for up to three months.