From the photo you provided it looks like you probably have lilac blight. We have been getting a lot of questions about it lately. 

You bush is not dead. You can prune off any individual dead branches but the most important thing you can do is clean up all fallen leaves and put them in your municipal compost. Do not compost them at home as your pile will not get hot enough to kill the fungi.

You can spray  it with a copper based fungicide in early spring before bud-break. But you have to to it early or it will be ineffective. 

Your lilac will probably come back fine next year. 

https://apps.extension.umn.edu/garden/diagnose/plant/deciduous/lilac/leavesblotches.html

https://extension.oregonstate.edu/news/how-recognize-treat-avoid-lilac-bacterial-blight#:~:text=If%20your%20lilac%20bush%20does,the%20buds%20begin%20to%20break.