This is a difficult one to answer, and is also the reason why I think Leylandii are so popular... The only climbers that are really vigorous enough to cover a medium to large tree tend to be deciduous themselves and so won't really do what you want them to do. You could try something like a clematis 'Montana', which will happily climb up through it and give a beautiful spread of flowers through early summer, but as I say it will drop its leaves.
The only thing I can think of that will cover it year round and not drop its leaves is the old Hedera ivy's, but a lot of people find these too invasive. It's worth remembering that all the weight added to the tree by the climber can stress the trees branches and even bring them down in strong winds.
The truth of the matter is that privacy is always a difficult thing to maintain, and there is no easy answer. Could you maybe add height to the back fence with a trellis and grow something up there instead? You may find this a lot easier to cover than a large chestnut tree!