How can I find where the dryer vent is broken or clogged?
You would need to run an inspection camera through it in order to identify exactly where the break or clog is.  Though, as the comments suggest, often it's easier and cheaper to just run a cleaning tool through to unclog it versus first running an inspection camera through.

I would probably only resort to an inspection camera if it became a chronic problem, which would suggest an issue in the pipe (sharp bend, collapse, etc.).  If it has only clogged once, and you've never cleaned it before, its quite possible it just clogged from normal use over a long time.