When I had issues with an old Mac I kept Activity Monitor running so that I could see if there was something in particular causing issues. Open that program up and see if you can see anything odd.
You also need to look for patterns, is the crash happening when you are running a particular program (Activity Monitor will help here because you will see if a particular program is taking a lot of power).
You say that you don't think it's overheating, but with the Macs we've had this kind of issue with that has always been the case. Does the Mac feel hot to touch?