Ableton Live does not have built in CDG support. But since Live 6 it lets you import Quicktime videos and play/manipulate them. There are some big limitations, but you can do the basics of KJ’ing no problem. The hard part is converting your CDG tracks to a video format Live can work with.
Here’s a quick guide on how to do that: How to covert karaoke files to video (Mac, Windows, Linux)
Pick one of the methods above (the one I can vouch for is kJams, see:Best Karaoke software for Mac) and try a test video. Once you’ve converted the video, import to Ableton Live by dragging it into the main window, where it says “Drop Files and Devices Here”.

If the file appears in the list of clips, you’re good. Now hit the “TAB” key to switch to Live’s “arrangement” video (more on this later) and drag the video into the main middle space (in this view, it isn’t marked “Drag Files… etc”).
The video should appear as another colored rectangle, and if you hit “Spacebar” in Live (or press play in the top row of buttons) a video window should appear and the track should start playing.

If you can’t drag the file into Live. Or if the video is garbled, that means the file is in a format that Live can’t work with. Go back and try one of the other tools in the guide above, or try tweaking that tool’s options. (The goal is a an h.264 video file in an .m4v wrapper. If the video plays in iTunes, it should work in Live.)
Note: Live 6 lets you work with video, but only Live 7 and later let you export your work. So you can perform with Live 6 but you will need Live 7 for making Youtube videos, etc. Since Live 6 was the last version with a stable crack for Mac (at the time of this writing) this is important to remember.
