The obvious answer is ivy - Hedera helix varieties, but they don't stop at 5 or 6 feet. It is though, if you are rigorous, possible to keep them as a 'living wall' against a fence or wall, at the height you want, by using a hedgetrimmer or shears to clip it over at least twice a year (usually in May and again late September), paying particular attention to the top parts to stop them getting too out of hand. A pair of secatuers on those more often will help with that. Attention should be paid to what happens near the base though - sometimes stray growths push forward at soil level and root as they go - they must be checked for and removed when clipping back.

What sort of fence are you trying to hide (eg timber panels, post and rail, chain link etc) and what condition is the fence in? Whose fence is it? Lots of things need to be considered especially if you are thinking of planting something that is going to rely on the fence for support.

If it's a timber fence and you are wanting to make a more pleasant view then painting might be a better option.

You might also want to mention what kind of sun exposure this fence will be getting. This will give an idea of which direction the primary growth will be going and the kind of soil conditions around the fence eg..moist or dry ,loose or dense? Which will help in choosing a successfull planting. Also how long is the fence? This will help determine at what intervals you might place the plantings to get adequate horizontal coverage. There are evergreens that you might consider too which could be planted on your side of the fence. Also, take heed in the comments of Ubanite if the fence is involved in a property border.

The fence is ours, have just put it up. It is timber panels and strong with posts set in concrete. The side I want to hide is in shade much of the day, the soil is dry and of poor quality. I don't want ivy.

You might find that the Arborvitae shrubs are very suitable to what you require. They are often used to hide fences, are very adaptable to different growing conditions, and can be as tall or wide as you will allow. How do I know? I have four of them hiding a thirty foot long seven foot high fence on my property.They grow very dense and give good privacy too.

I think I'd go with painting it a nice colour - there are some great shades around these days - that could act as a foil for other planting in a border. If it's new panels and you're only looking for something to grow up to 6ft then, presumably, privacy isn't the issue. Also, planting a hedge against the fence could lead to maintenance problems in the future.

I looked at your photos (stunning setting!) and you could pick up the colours of the flints in your garden walls or the cream of the house walls.

Have you looked at the Parthenocussus?
Not evergreen but would certainly cover the fence and is self clinging,youcanfind photos on Goy, go to P below and scroll throught he photosor put the name in the search box, goes by the names ofboston ivy or virginia creeper.

looking at C for climbers below may show more ideas for you

I made the assumption that the fence was a state, and that's why you wanted to cover it up. Why not just paint it a nice colour and plant freestanding shrubs in front of it (if there's a border). Otherwise, Cotoneaster horizontalis, though it won't stay neatly like a green wall against the fence.

There are a few climbing roses that will grow in that position - I had a lovely gold one but sorry don't remember what it was.Avoid the ones like Golden showers though as they would probably get too big. I also had a small flowered clematis growing with it. Not dense coverage and nothing much in the winter, but very pretty when both in flower together.
The cotoniaster is a good idea - although there are no leaves in winter there should be lots of red berries and lots of interested birds while they are still there!
You can also train pyracantha - I did one on a fence under fairly deep shade from a large sycamore and it grew quite well. I just used canes to weave back the new outward facing growth and it worked very well. You would need several though.

Garrya elliptica 'James Roof', Hydrangea seemanii, H. petiolaris, Schizophragma hydrangeoides, Chaenomeles.

Thanks for all your useful comments. I like the idea of cotoneaster.