Lilac tree.

mikebez1

I have a lilac tree and twenty or so thick suckers. I have chopped half down to two feet. I really want to kill it and roots so as no repeat suckers occur. Anyone have experience of this. It’s too much to dig up. 

Blue Onion

If it's in the yard, cut it all down to an inch below soil level.. and mow over it for a year.  This will take care of suckers.  If it's in your boarder.. cut it all down and use a hoe to easily 'chop' suckers off at soil level wherever they grow up.  You might get a few odd ones up next year, but just keep removing them.