| Sounds like you bought my house. We are on 1/8 acre, the house behind us is at a higher elevation. Our municipality only allows 4 1/2 foot fences. We built a berm at the back of our property, put a fence on top, installed a 1 foot high lannonstone wall about 3 feet in front of the fence, back filled that with soil and planted a row of purple leaf sandcherries on top, an elevated planting bed or so to speak. I'm letting the sandcherries grow up into a tree form, I have four of them lined up along the fence with an evergreen arborvitae in the corner for contrast. While the sandcherries won't give us total privacy, it's enough that when they grow they will obscure the view from the neighbors kitchen window to our patio, at least during summer months. And, they add a nice touch of color to our yard. I have another sandcherry that I cut down every year to keep it's shrub form, it blocks the view of my garbage can (at the corner of the garage). Our patio is next to our house. We also planted a popple, aka quaking aspen tree closer to our patio. There are pros and cons to this tree, but it's definitely fast growing. You aren't supposed to plant it close to a house simply because as they age they tend to hollow, but we lop the top off every year to a managable height so it grows wider than taller limiting the risk to our house. They also tend to send shoots around the yard, but easy enough to mow over. The tree gives us privacy from the house behind us, but only shades the area next to our house/patio, so I still have sun in the back part of my yard for gardening purposes. The back of my house faces west. If I could go back, I'd plant a serviceberry "autumn brilliance" in place of the popple - the roots are more shallow and not invasive. We originally had Techny arborvitae along our back fenceline, but as slow growers they filled out in width more than height, and tend to take up too much room, though I did like the fact that they were evergreen. Another thought is Canadian Hemlock for an evergreen privacy screen. While this is technically a tree, it can be easily shaped into a hedge form for total privacy, though I have no personal experience. I'm thinking of planting one in another area to screen another view. Between my fence and patio I left about 3 feet planting space, and planted a staggered row of privets, the Cheyenne variety, to provide privacy from the neighbors to my north. The first year they were twigs, this year they were nice and full and almost reach the top of my 4 1/2 foot fence. Next year they will exceed the height of my fence and hopefully in two years we'll have privacy. They can be shaped into a hedge form. For every 12 inches they grow, cut them back 6 for full growth top to bottom, otherwise they tend to get leggy. There are pros and cons to privets as well, do the research before planting. A fast fix for privacy is the annual morning glory on a trellis. We did this for a screen this year, and I couldn't have been happier. William Baffin rose is a rambler, grows into a high arching form if you have a nice sunny area maybe in a corner of your yard - the key to making your yard feel more private is by adding height to it. A nice patio tree is the Rose of Sharon. Place trees strategically any where you want to block a view. Whatever you do, DO NOT plant the hyped up fast growing Cottonwood trees. They will destroy sewer pipes, foundation, etc. and your neighbors will soon be your enemies. LOL. 
Another view:
 "Cheyenne' Privet hedge first year:
 Privet hedge second year:
 Morning glories on trellis:
 Popple, 2nd year?
 This is what we started with:
 Not my yard, but nice example of mature sandcherries in trimmed up to tree form: 
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