Old Answer (July 2016):
You can't directly debug Chrome for iOS due to restrictions on the published WKWebView apps, but there are a few options already discussed in other SO threads:
If you can reproduce the issue in Safari as well, then use Remote Debugging with Safari Web Inspector. This would be the easiest approach.
WeInRe allows some simple debugging, using a simple client-server model. It's not fully featured, but it may well be enough for your problem. See instructions on set up here.
You could try and create a simple WKWebView browser app (some instructions here), or look for an existing one on GitHub. Since Chrome uses the same rendering engine, you could debug using that, as it will be close to what Chrome produces.
There's a "bug" opened up for WebKit: Allow Web Inspector usage for release builds of WKWebView. If and when we get an API to WKWebView, Chrome for iOS would be debuggable.
Update January 2018:
Since my answer back in 2016, some work has been done to improve things.
There is a recent project called RemoteDebug iOS WebKit Adapter, by some of the Microsoft team. It's an adapter that handles the API differences between Webkit Remote Debugging Protocol and Chrome Debugging Protocol, and this allows you to debug iOS WebViews in any app that supports the protocol - Chrome DevTools, VS Code etc.
Check out the getting started guide in the repo, which is quite detailed.
If you are interesting, you can read up on the background and architecture here.