If you are saying the container you want to plant into is only 4 inches deep, then no climber will grow from it. You'd need something around 2 feet deep and a minimum of 12 inches wide for a climber to get any size at all, I'm afraid.

If you can provide a pot large enough, Hydrangea anomala will cope with the shade, but it is self clinging after the first couple of years. Otherwise, something like Fatshedera lizei, gets about 4 feet, needs support, evergreen, should also cope and will tolerate being in a pot for longer.

I 'think' Alan meant to say 'plot' but four inches certainly isn't anything like deep enough...