The components needed are: Potting soil, a bucket, water, fertilizer (liquid 4,1,1 or eqivalent), quart size baggies (2"x4"x12" plastic bags can be ordered from Amazon, Ebay, or Sam's Club), some string like knitting yarn, a roll of aluminum foil and a cup that will hold about 1 cup of potting mix (about the size of a tennis ball).
Per the instructions for the fertilizer selected, add the appropriate amount of fertilizer mix to a gallon of water. Put about three gallons of potting soil in the bucket and add the gallon of fertilizer mix to the potting mix.
Place a tennis ball size of the mix into a plastic bag. Gently place the tip of the blackberry cane into the bag and tie off the top of the bag around the cane. As long as the cane is in contact with the soil, roots will be starting to grow after a few days.
The bag and cane can be suspended from the trellis while the roots are developing. I've learned by experience that it is best to ensure the cane extending out of the plastic bag is oriented so there will be at least a 16 to 18 inch straight section. This can be done by suspending the bag 16 to 18 inches below the trellis as it being tied.
In order to minimize the temperature in the bag, wrap the bag with a piece of aluminum foil.
Either before or after wrapping the bag with the foil, make a small hole in the bottom of the bag (a pen or pencil will work for this). This will allow rain water that runs down the cain and seeps into the bag to drain.
After a week there should be roots beginning to show in the bag. Wait about another 10 to 14 days and then prune the cane about 16 inches from the bag. Transplant the new young plant to a one gallon pot. The roots can be positioned in any orientation in order to get the cain completely verticle. Ensure all of the root ball is covered with potting soil. Water the new transplant thoroughly.
Another 10 days and the roots should be starting to fill the bigger pot.
If the newly potted plant is placed on a surface where the roots cannot find soil within a few inches, they will self prune and both root and vine growth will slow down as the plant becomes root bound.
If desired, the propagation can be planted directly into the ground instead of the pot. If you are planting into a clay soil ensure there is no chance of water pooling around the roots.