Sorry to hear this.* Unfortunately your lilac was probably never pruned properly and that's why it's ugly.* The proper way to prune a tired and overgrown lilac is to cut one third of the oldest trunks flush to the ground.* Do one third each year for 3 years and the newer shoots will then begin to bloom for you.* Lilacs will send up shoots to renew themselves.*
If your lilac doesn't bloom or bloom well it could be from improper pruning.* They set their buds for the following years blooms about 2 weeks after the finish of bloom.* If you prune after that, you are pruning off the buds for next year.* Lilacs also don't like lots of nitrogen fertilizer.* If yours is in the lawn and the lawn gets fertilized, that will cause lots of leafy growth at the expense of flowers.* They also prefer a more alkaline soil, so if your soil is on the acid side, adding a cup of lime and mixing it into the soil around the base will help.* They need full sun.
If you still want to remove it, your only choice would be to cut the trunks nearly flush to the ground and dig out the main roots.* Any sprouts in the lawn will eventually give up with continued mowing or they can be dug out as they appear.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.