Hi,
Thanks for your question.

You have two options to permanently remove lilacs. One option is to exhaust the root system by repeatedly cutting the bushes back. However, this could take a couple of years, especially if new suckers are allowed to leaf out and renew the roots.

Another option is to apply a systemic herbicide that will travel down and kill the roots. 

September is actually a good time to apply an herbicide for two reasons. First, the daytime and nighttime temperatures are still warm enough for the chemical action to be effective. Second, perennial plants are busy transporting nutrients down to their roots in preparation for winter, and you need that transport system to carry the herbicide to its target. 

There are two kinds of herbicides you could consider. Glyphosate is a non-selective herbicide and is used in brands such as RoundUp. Triclopyr is a selective broadleaf herbicide. In either case, shield any desirable plants nearby. Cut the plant down to one inch above ground level and paint the freshly cut stumps with herbicide using a foam brush. Spray foliage of suckers with herbicide.

Read the instructions for use before you buy an herbicide to make sure you can meet the requirements for temperature and timing.

If you do this removal project in the spring, keep in mind that you'll need to apply herbicide to a plant that is actively growing under specific temperature conditions.

Good luck with your lilac removal project!