How to debug an OS X crash?

Start with log files. You have a program in your Applications > Utilities folder called Console; Fire that up, and it will show you a collection of log files - you should inspect these to see if there is anything obvious showing up at the time of your crashes. Once you have discovered something that looks odd, or consistent across your crashes, you can google it or ask again here.

Apple provide a help page on If your Mac restart because of problem... And if you want to actually attempt to debug a kernel crash then another page on Kernel Core Dumps which explains that you'll need to setup a debug server and also configure your crashing machine to dump core and/or debug to that server machine.

The other possibility is that whilst the GUI may be unresponsive the kernel may not have crashed so you might be able to connect remotely using ssh or remote desktop to attempt to ascertain the problem.