Thanks for cutting down the chokecherry tree, and starting the process of dealing with it.  Without herbicides your best bet is to try and continually cut off new branches coming out, and any suckers coming off roots.  Theoretically if you do this as you see new sprouts during the summer in a few years it should exhaust the tree.  I often hear from people that take this route that they have a hard time keeping up on it, and they continually get more and more sprouts.

Herbicides are very effective. What herbicide you use and how you apply it depends on the site and vegetation in the area, and what you are comfortable with and have access too.  Typically I send people who cut down the tree and now have resprouts on the route of doing a foliar application. A product containing Glyphosate, 2,4-D, or Triclopyr will work. Triclopyr is really good at killing trees.  With any of these you want to make sure the herbicide only gets on the vegetation you want to kill. The only exception is 2,4-D and Triclopyr won't kill grasses. We have a guide on doing this work, that shows some other application methods too, find it here. 

I don't usually recommend the cut stump, frill, injection, or basal bark for people with one tree because you use concentrated herbicides for those treatments which means you will buy a lot more than you will ever use. If that is desired though you might either find a contractor to do that, most tree care/landscaping companies, so you don't have to deal with storage/disposal of the concentrate.   

Let me know if you have more questions and good luck.