Do you know what type of blackberry you're growing? Some varieties are naturally quite tart. They're usually used for jams and pies (because you can add sugar to make them more palatable) instead of eating fresh. You can improve the general quality and flavor of the fruit by keeping the soil around the plants well covered with organic mulch (such as compost, straw, or shredded leaves). Mulch feeds the soil, which helps strengthen the plant and thus improves the fruit quality. Mulch also helps prevent rapid drying of the soil, which can stress the plant and cause irregular, shriveled, or hard berries. In the end, you can't overcome the heritage of a naturally tart variety, but you can get it to produce its best possible fruit. If you don't like the fruit you have even after trying the above recommendation, replace it with a sweeter-tasting variety.