I think you are looking for two things: short term and long term. So, what about hops for short term? it is a perennial, it grows in zone 3 so you would be fine, and you can often find some from another gardner for free. comes in a lot of varieties. It is a rapid grower, and covers chain link fast.
It won't really help a heck of a lot in winter, but..that gets you to long term: planting evergreens and woody bushes.
Also what about a raspberry hedge? think layered, you could do hops or wild clematis on the fence itself, bushes, evergreens, raspberries, forsythia, in front of the vines on the fence. OH trumpet vine too.
Maybe start with the fast growers, plant the longer term guys like grape vines, trumpet vines, etc.....at the same time.
You could do a section at a time as your $ allows....
And I love the climbing roses Mumsy has... are you able to plant on the street side of the fence? you could protect stuff from goats on that side and have goat resistent (really is there such a thing?) on the inside.