There's an orchard farm near here which has a 2m high hedge of single beech stems planted along the road side. It looks lovely all year. Theyv'e planted stems at 6 to 9 inch intervals (15cm to 23cms) in a straight line and let it grow about 12"/30cms in depth.
You could do the same but it's best to wait till next autumn as the cheapest plants are sold as bare root whips and they'll establish quickly without needing copious watering. Dig a trench along the fence and about a foot (30cms) away from it and add plenty of well rotted garden compost or manure then put in each stem burying it to just above where the roots break. You can add bonemeal to help root formation or mycrorhizal fungi following the instructions on the packet. Cut each stem back to about 12 inches or 30 cms, water well and leave to grow over tthe winter.
If your soil is heavy or gets wet in winter, use hornbeam instead of beech.
Another alternative is simply to attach a strip of trellis along the top of the fence to add instant height. You can then stretch horizontal wires at 12"/30cms intervals along the fence and plant climbers such as roses, honeysuckle and/or clematis or wall shrubs such as pyracantha which is evergreen and provides shelter and food for birds and insects as well as thorns to deter climbing hoomans.