You need the pulp to make blackberry jam. The juice will give you a wonderful jelly, but there's nothing like jam/preserves.
When doing a small amount, I use a wire strainer that closely resembles the spacing of window screen. Actually, I remember as a kid watching a nearby farmer's wife "squish" hers through a frame of window screen into a container like a plastic dishpan. She used something that looked like a flat wooden paddle. It was quite a fine screen, but it let the pulp pass through while still holding back most of the seeds.