It doesn't sound to me like it's actual weeds, but called weeds as in it is overgrown yard along the fence. The issue is not being trimmed, not that there's actual weed growth in grass there.Weed whack your side of the chain link fence. After doing so, put down some pre-emergent herbicide pellets...Home Depot and Lowes sells it. That will prevent weed seeds from germinating in your yard going forward. Reapply every 3-4 months.