Blackberries bear fruit on canes that grew the previous year. Once the canes finish bearing for the season, cut them off at the base. Remove only the canes that bore fruit that year. They will be woodier and have fruit stalks on the canes. New canes that started growth that spring will take their place as the bearing canes the following year.
In winter, when the new canes are dormant, shorten them by about one-third to encourage branching of fruiting shoots lower down on the canes in spring. Blackberries do best with lots of organic mulch: compost, shredded leaves, hay, etc. Be sure to keep the soil moisture constant.