The process of testing the high voltage capacitor is basically the same as testing the other main components in your microwave. You will first need to remove the terminals from the component before testing them with your multimeter and checking for continuity.
The issues with a capacitor necessitate similar action to that of a failed magnetron. It is a relatively inexpensive part (depending on the model and manufacturer of the device), but it could still be easier to simply pick up a new microwave.
If you know exactly what kind of capacitor you need and how to replace it yourself, you can order one online for a low price.