This could well be verticillium wilt, which unfortunately attacks lilac trees, although rarely. It’s caused by a soil inhabiting fungus which clogs the plants vascular system. Wood under the bark may exhibit discolored streaks or bands. Confirmed ID is only possible through laboratory tests.
There is no chemical cure. Infected trees should be watered throughout the growing season. High nitrogen fertilizers increase wilt severity; applying a 10-10-10 may help. Remove all dead branches, pruning back to wood showing no streaking. If you replace your tree, choose a species that’s not subject to verticillium. Particularly susceptible plants—ones to avoid—include maples, smoke bush, sumac, barberry, Catalpa, redbud, and roses.