Who cares if you can amend the soil, you dont have any sun... Are you going to buy some grow lights to hang over your patch?

To answer your question you can till in a lot of sand to loosen the soil but adding organic matter will help the most. In a container I would use 50 % good black dirt/topsoil mixed with 30% peat and 20% compost (or 15% vermiculite/perilite and 5% aged manure). You can build a raised bed using standard pine 2 x 12's that will last for many years, or use cedar and it will last a few decades.

I have raspberries growing in 18 gal containers just east of a 6 ft. fence. They get sun until about 2-3 o'clock, depending on the season, and of course more as they get taller. They seem to do fine. Like you, I have shade issues from a gigantic rebuild to the south (and my own house to the east) so I have to be choosy about what will get the most sun and when during the day it gets it. It's also generally cool here, just a couple of blocks from the back bay.

I grow the plants in containers because I have almost the exact opposite soil problem: dune sand. If I did manage to get the raspberries going in the ground they'd soon be everywhere, so elevated containers were my solution. They share a netted enclosure with my containerized blueberries.

For the soil mixture I use more or less a poor-mans 5-1-1- mixture made with small pathway bark, peat moss and pearlite and a little dolomite. This mixture drains pretty well but does require constant added nutrition because there's not much in it. I originally start with a little Osmocote mixed in but it doesn't last forever.