Whatever you get you'll have to make sure it's cane hardy in zone 5 or it'll die back each winter and you'll start from scratch each spring. My New Dawn had very little dieback this year. I have others that did well but they are not usually available in garden centers or nurserys.
The average climber easily available will suffer a lot of winter dieback.

You haven't indicated the dimensions of your privacy fence. How long do you want the canes to be?

For a good hardy climber, I think an easy one to find would be Ramblin Red. A bit harder to find locally but good climbers that have done well in my 4B zone: William Baffin & John Cabot. I have not yet tried Dortmund but have read it is a good hardy climber. The Lioness

Roses are an investment. They are not a one time plant. Make an effort to buy the best you can, even from an online supplier.
One important issue to consider is how much sun will the rose actually get. From your description, it looks like it will live in part shade.
Some people recommend Zephirine Drouhin for shaded areas, others reject this statement.
Make a search on "Help me find roses" and check the websites of some recommended nurseries.

I am trying to ascertain if the OP is from Austria given the screen name and if so, their zones will not be USDA zones.

Do you spray?

As far as size, I have Westerland, Awakening, Ramblin Red, Climbing James Galway, as well as some others that get used as climbers like Darwin's Enigma. Galway gets the biggest and longest canes and it's fairly thornless, so that's nice.

But far and away the most vigorous, biggest, and fastest growing is a rose called Pleasantrie, which wasn't even supposed to be a climber. It dwarfs all of the others by far, with 20 foot canes in its second year. In fact it's way too big and I may have to get rid of it. In comparison, the others aren't climbers at all.

I would suggest Climbing 4th Of July. I have one that gets morning shade and afternoon sun. It winter killed very little last winter (we had a fairly rough wintter) and it is good repeat bloomer. It is easy to find in a nursery. Mine is going to end up being a huge rose.

That Fourth of July is a terrific rose in a lot of places.
It's also great in SoCalif, blooming heavily, and with no disease.
If only I had the space, I'd grow it as a big fountaining shrub -- I'm told it's spectacular used that way.

Jeri