Prep. Set your oven to 350°F. Then, place a rack in the middle of your oven. To prepare the Bundt pan, you can either grease it with shortening and a sprinkle of flour or use a non-stick cooking spray that contains flour.
Cream butter and sugar. In a large mixing bowl, use an electric mixer to cream the butter and sugar together at a medium-high speed until it becomes light and fluffy for about 3 to 5 minutes. Then, include the eggs one at a time, mixing well after each addition until it is fully mixed.
Dry ingredients. In a separate bowl, combine the dry ingredients (the flour, salt, and baking powder). Add the dry ingredients to the butter mixture at a low speed, 1/3 at a time, alternating with the milk.
Add fruits and flavors. Gently fold in the blackberries, lemon zest, and lemon juice by hand, making sure they are evenly distributed without crushing the berries.
Bake. Pour the batter into the prepared pan, smoothing the top with a spatula. Bake for 45 to 65 minutes, or until a toothpick inserted into the center of the cake comes out clean or with just a few moist crumbs attached.
Cool and serve. Let the cake cool in the pan on a wire rack for around 15 minutes. After that, gently flip the cake onto the rack to let it cool down entirely. Once it's cooled, sprinkle some powdered sugar on top if you want, and serve it as you like, maybe with whipped cream or some fresh fruits on the side.