Consult your local agriculture extension agent or berry farmer to find out the types of raspberry bushes that are best suited for your area.
Line the bottom of a large container, at least 5 gallons, with gravel or broken clay pieces for added drainage. Dwarf raspberry plants should do well in a 5-gallon pot, but larger varieties would do better in a much bigger pot.
Place the container in a section of your balcony that receives the most sunlight, and position it against a wall or balcony rail so it doesn't crowd the walking space. Containers are much more difficult to move into place when they are full of soil.
Fill the container to within 1 inch of the top with sandy loam potting soil and organic compost. Raspberries prefer soil that drains well; mixing sand into potting soil improves drainage.
Plant the raspberry to the same depth as its nursery container, and loosely pack soil in a mound around the base. The mounded soil will encourage water to drain away from the plant.
Water often without over-watering; raspberries need 1 to 2 inches of water per week. The soil should form a slightly crumbly ball when squeezed.
Train the raspberry plant to grow against the balcony railing for support, or install a trellis onto the balcony. If the raspberry bush sits in a corner of the balcony, run a few strands of galvanized wire over the bush to provide support as it grows.
Prune the raspberry bush regularly to control the spread of the plant and keep the canes from tangling. A well-pruned bush produces better fruit. The amount of pruning required varies greatly with each raspberry variety, but may require removing lateral shoots and trimming the tips.
Cut the raspberry canes to ground level or prune back the canes after you harvest all the berries. Add a layer of mulch to the soil for insulation. If everbearing raspberries are cut down completely, the bush will come back the following spring with new canes and produce only one harvest in the fall. If the canes are simply pruned, the old ones will produce fruit in spring, and new shoots will produce a second harvest in the fall.