Do you have a list of the files you have modified in the core theme?
Provided that you know what files you have modified, you can make a list, and copy those files into a child-theme folder (make sure to include the style.css file along with these.
If you do not have a list of the files you have modified somewhere on hand, an alternative is finding an unmodified copy of your theme's current version on your hard-drive, and looking at the last modified timestamp for all of the files, and comparing that with your local copy's last modified date... Make a list of the files that do not have matching timestamps, and then copy those to a child-theme directory along with your style.css file from the parent theme. ( This is how I did it in a past project after learning what a child-theme was for the first time. )
Next open up the child-theme's style.css and add the template tag that points to your parent theme as directed on the Codex Child Themes page.
Upload the child-theme and activate.
Make a backup copy of your current parent theme!
Finally, upgrade the theme.
I would definitely advise you do this on a development server or test site first, that way if you miss a file, the design goofs up, or you get the WSOD, you're not affecting the client's live website.
I hope this helps!