There are many ways. I start with a thurough visual inspection including pulling the limit to check for missing crimp rings if it's a 2 piece Hx.
I then start the unit(with blower door on) and watch it fire. Closesly watch the flame pattern. Once the blower turns on the flame pattern should not distort from it's shape prior to fan operation. If it does this would warrant further exploration.
If worse come to worse you can check CO output at the flue outside or pull the flue off and check it right at the unit. Some have differing opinions on what a proper Co output should be though I look for no more than 125ppm. If you hit 400ppm you have a problem, I have seen 600ppm. This clearly shows a combustion issue though it may not be a failed heat exchanger. It indicates improper combustion or impingement of the flame. It could be something as simple as lack of combustion air or even improper gas pressures. However, the high Co reading will tell you if you need to investigate deeper into the system.
While it takes some time I have pulled heat exchangers if I feel there is something there that needed attention. Everytime I have chose to go to that extent I have always ended up finding something and condemning the unit.
It all really does come down to experience. The more furnaces you see, the easier it'll be for your eye to catch something that isn't quite right. I for instance have found that even though Co is odorless, there is a specific odor to a badly failed heat exchanger from the improper combustion, once you smell it you'll never forget it. Just pay close attention to every unit and get practiced up, you'll find your own little tricks to finding problems.
There are 3 ways to do anything in life; Good, Fast, Slow: You can pick any 2.