I'm new to gardening and looking for a some climbers to put in my North facing lightwell. It is very sheltered and gets a little bit of sun in the summer evenings. I have bought two very large containers (60x60x60cm) to put the plants in and erected some trellis for them to grow up. Ideally I would like something that flowers but my main priority is to ensure good coverage. I will be getting one trachelospermum jasminoides recommended in other posts but would appreciate any advice on something else to get.
The problem with Trachelospermum is it likes not only sheltered situations, but full sun, or as much as it can get, so I'll be interested to hear how well that does in the circumstances you describe.
Evergreen, low light conditions most of the year, and the constriction of a pot makes for limited choices. Have a look at Fatshedera lizei - this isn't a climber as such, it will need tying in to your trellis or other support to keep it upright as it grows, but it tolerates low light conditions, has attractive architectural leaves and is evergreen, but does not flower. There are variegated versions available (F. lizei 'Annemieke', F. 'Aurea' or 'Variegata'), but the variegation may not be noticeable under low light conditions. Otherwise, Hydrangea anomala (syn. petiolaris) should cope. Note that all these will need regular attention paid to watering because they are in containers, and that the containers should have drainage.