Start by telling us where you are and your USDA growing zone.

Check with the fence’s owner to see if they would permit you to paint the fence a darker color so it blends. A dark green will work best if you can get permission.

You don’t need enormous plants like Green Giants which grow hugely tall and wide. The woodland behind the fence will make a beautiful borrowed landscape as is, so planting things that will be tall enough to hide the fence will be all that is needed.

Are you looking only at evergreens?

How much space are you willing to devote to this coming out from the fence? Having more than one layer of plants will give you better coverage sooner as well as allowing a mix of evergreen and deciduous plants for more variety and for flowers if that is what you want.

I had a difficult time finding photos of appealing mixed shrub borders, so will link a thread here on Houzz/Gardenweb where I posted mine at planting and 4 years down later. Whether you can use any of the same plants will depend on your location as well as the aesthetic you have in mind. My posts start about half way down the thread.

https://www.gardenweb.com/discussions/3113384/shrubs-for-privacy-along-a-fence

A few general pieces of advice:

-Pay attention to watering the first season or two. Otherwise you may lose plants.

-Don’t use landscape fabric. That is fine under paths but in a garden interferes with movement of air, moisture, and nutrients that plants need. It won’t prevent weeds and makes removal of those that grow more difficult. Use a good layer of an organic mulch such as shredded bark, and renew it as needed to keep a layer of 3” or so. If you want to, put a layer of a few sheets of newspaper under the mulch to reduce weeds initially.

-Edge the bed with something that goes below the surface with a continuous edge, either a deep V cut or a buried strip edging with a mowing strip of masonry set to the bed’s surface. This will keep out lawn plants to reduce weeding.
-There is no way to avoid weeding, but if you keep on top of it and don’t let things get out of control, it need not be onerous. Use a rake to scuffle up the mulch surface once a week on a sunny day. Any weeds that are still green the next day remove by hand.

- When you plant, don’t plant for full coverage immediately. If you do, plants will be way overgrown within a few short years. Note in my photos on the thread linked above how fast plants grew in just 4 years. Allow the needed space for growth to maturity. Plant tags typically give the 5 or perhaps 10 year growth, so you will need to look up actual mature sizes on a reliable site such as the Missouri Botanical Garden’s Plantfinder.you can either use it to search for plants with particularly qualities such as size, bloom, evergreen or not (and you can leave some factors blank) or type in particular plant names to get info on those plants.

Seeing a piece of the fence line is not enough information to make a plan. It looks like light conditions vary because of nearby tree canopies at some places. You'd need to draw out, to scale, the fence line, and plot out what is shade and what is sun. Then you could determine the plants that can tolerate the conditions. Likely, you'd want to place some of the larger material (such as Nellie Stevens) where there are not tree canopies above, and place some shorter, shade tolerant material where canopies exist. No one knows your location (nearest large city), so impossible to recommend plants.

To minimize weeds, you need to get rid of them (with glyphosate) before planting anything, and then add a thick layer of mulch. For at least a couple of years, it would be good to treat the area with Preen on a schedule (after planting.)

I am assuming you live in approx. zone 7 based on your plant likes. Here are some ideas using Needlepoint holly, variegated aucuba & Japanese Forest grass:

Sorry- yes we are zone 7 and the area gets light for most of the day.

No need to use gyloohosphate. . Lay thick newspaper (6 sheets) down along the proposed bed, wet it and cover with mulch. Wood chips are usually delivered free by tree companies. When you plant just cut through the paper. You will have a mulched, weed free bed with superior soil within a year. New weeds will be minimal.

You could cover the white fence with bamboo fencing which comes in rolls at Home Depot or Walmart. It lasts about as long as it will take to fill in your plant screen.