Get a string trimmer or some shears and cut all that overgrowth down so you can see what you’re working with.
Dig up the roots with a garden fork or spade. Just trimming the tops won’t cut it – you gotta get those roots out.
Spray a glyphosate-based herbicide on any remaining plants. Follow the directions, do it on a calm, dry day, and make sure it doesn’t drift to plants you wanna keep. This stuff kills everything it touches.
Lay down some landscape fabric or plastic edging along the fence line and cover it with mulch. This stops new growth from popping up.
Regularly check the area and kill any new growth ASAP. A quick herbicide spray on small patches will keep things in check.
If you don’t like chemicals, you can pour boiling water or use a vinegar solution (mix vinegar with salt and dish soap) to kill plants. These might need more frequent applications but work well too.