Testing Electrolytic Capacitor
The best way of testing a capacitor will depends to the type of capacitor you are going to test. Electrolytic capacitors that are leaky are not so common. You can take each capacitor out on the circuit and test it using a cap checker or using your VOM, but in-circuit testing of the capacitor is quicker and save a lot of time.
Do a visual observation and inspection to the capacitor and see if there are any electrolytic capacitor are bulging (they are leaky and commonly get hot), or physically leaking (corroded terminals). Next, if the unit will power up, search for indications of an open filter caps, hum bars in the display/picture, hum in audio, flickering displays, low B+ but nothing gets hot, etc.
If you have an noticeable fault, like a lowered vertical scan on a TV set or monitor for example, to locate the capacitor that is starting to open up, you can bridge each of them with another capacitor, one at a time and see if it fixes the problem. (Experience has taught me that bad electrolytics will not -usually- kill vertical sweep completely.) In a TV set that is several years old or more, there could be more than one cap dried out (open). Check them all.
As mentioned before, leaky caps are actually quite rare, but it does happen. Shorted tantalums and electrolytics capacitors in a power supplies can actually explode. Obviously, leaky caps must be taken out from the circuit to replace them for test reasons.
Most of the other types of small capacitors: mylar, disc ceramic, etc. are pretty rugged. It is rare indeed to find them bad. It happens just often enough to keep a tech humble. But I also encounters that this small caps are the culprit, and the problem is that they are get leaky.