Electronic ignitions are pretty easy to access and pretty easy to repair. On the majority of bikes they are comprised of only 4 components: the CDI, the ignition coil, the pickup coil and the trigger coil.
The trigger coil tells the CDI when to fire the spark, as a magnet connected to the flywheel passes by. The electricity for the spark may come from either the battery or a source coil, depending on the type of bike.
The way to test these parts is to measure their resistance in ohms. This is when your repair manual comes in handy, if you are lucky you will get detailed instructions on how to test across which leads, and what the readings should be. Sometimes it is impossible to tell, and your only option is to swap in a new part and see if that improves the situation.
This video here demonstrates the process:
A couple important things to remember:
1. Just because you found a defective part, doesn't mean it is the only defective part. Sometimes when a part fails, it takes another one out with it due to the unusually high voltages that are produced. 2. Having a good relationship with a forum can come in handy here. CDI boxes are expensive, if you suspect that yours is bad, you can mail it to another forum member and he can plug it into his bike and test it. I have done this many times and saved quite a bit of money in replacement parts. 3. What at all possible, try to test the continuity of your parts in an environment which is as close to 70deg F as possible.
So, just take it easy, and go about it methodically. Your repair manual will help out a lot.