Clematis are 'scramblers' rather than 'climbers' and so will happily scramble over your fence. In the first instance you will probably have to encorage it by tying stems to the fence but, once it gets going it will be quite delighted to do it's own thing.

If you don't mind the way it looks, the easiest solution is to use white clematis mesh - comes in a roll over here, must be available over there. Fix that to the picket fence, then just let the clematis do its thing.