Your biggest problem is shade. Not many vegetables will thrive in a shady spot, so I would concentrate on interesting expensive salads etc.
The problem with shade is that plants will look as though they are growing, but your results will be disappointing. I have been blighted with this problem for years, and would recommend:
Salad mixes - especially rocket, mizuna and other highly flavoured baby leaf salads. (as long as you like them of course).
I would also put in some herbs that you like to cook with, and those that are much nicer fresh. Rosemary, thyme, french tarragon. basil and coriander in summer.
Chives will survive in shade.
You could create a small raised bed for salad and herbs, which you can then put compost into for better drainage than soil (depends on your soil type whether you need to)
In addition, you can grow some berried fruit successfully in partial shade. Blackberries, raspberries, gooseberries will all do ok. You can apparently grow a morello cherry tree on a north facing wall - this would look nice trained into a fan or espalier?
I have never tried potatoes in shade, but a dustbin-type thing has to be worth it with some first/second earlies.
A book I would recommend is called 'Urban eden' by adam caplin which gives lots of great tips for growing edibles in a limited space and coping with aspects like shade and poor soil.