The GrippleTensioner Kit will certainly support your clematis very well indeed. I use this kit a lot in the Kitchen Garden to support espalier apples and trained soft fruit. We also use it to train Jasmine in other parts of the garden. It is one of my personal favourite Harrod products and I recommend it.
It is amazingly easy to use and very effective. You will need something to attach the wire to at each end of the area you wish to cover. Vine eyes are only suitable for walls, rather than fences. The tensioners pull the wires tight once they have been wrapped around posts orthrough hooks. In the garden we have timber posts at each end of the training area which the wires are then wrapped around. Alternatively you could put some hooks into your fence and attach the wires to them.
I hope this is helpful, but please contact me again if you require any further information.