Solanum crispum‘Glasnevin’ is semi-evergreen where I live but grows very quickly to almost shrub like proportions, with attractive purple flowers. There's also a more delicate variety with white flowers, Solanum laxum "Album", again semi-evergreen, which means you'll still have a few green leaves over winter. Like @Fairygirl , I also thought that Clematis armandii might be suitable, provided your garden is sheltered and you don't get extreme winters. It flowers in Spring and is evergreen.
I would definitely not paint the fence black or dark as it would enhance the entire lot. If it's a lighter colour the eye would be drawn to that rather than a big dark area.
I would be aware of the Leylandii roots too, you might find it difficult to dig holes for your new plants. Good prep is essential plus water throughout this growing season and in future years. You may find the leylandii is just too competitive for other plants. It will take all the water and nutrients. Not what you want to hear I know.
I have worked as a Gardener for 24 years. My latest garden is a new build garden on heavy clay.
A lot of climbers would need a little help up the fence, but will then happily grow along the leylandii. Finding something to do that which doesn't then become difficult to manage could be a challenge.
One option would be annual climbers - nasturtiums, runner beans or mashua come to mind (all with the added benefit of being edible)...
Absolutely agree @Plantminded , although I'd say that's nearer white than grey
Any dark colour will make the fence recede - whether it's black, brown or green. Anything lighter or brighter will draw the eye. It would certainly help to have some additional fence/trellis though, also painted, as that will help block those dead trunks until any climbers get going, because that won't be instant. It'll take several years for climbers to successfully screen those.
It's a place where beautiful isn't enough of a word....
I live in west central Scotland - not where that photo is...
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I think @Plantminded ’s is an excellent example of how well it works, my garden is a little young but this is how it looked in peak summer:
Any dark colour will make the fence recede - whether it's black, brown or green. Anything lighter or brighter will draw the eye.
It would certainly help to have some additional fence/trellis though, also painted, as that will help block those dead trunks until any climbers get going, because that won't be instant. It'll take several years for climbers to successfully screen those.
I live in west central Scotland - not where that photo is...