This is what I have learned so far, I'm happy to be corrected on any of this.
A test with a multimeter is an indicative test but not a proper test.
1. On a multimeter you should be able to set it ohms range (resistance measuring) and watch the resistance go up and down by swapping leads to charge and discharge the capacitor. The capacitor may also discharge by itself in circuit or by normal leakage. What range you select on the multimeter depends on the capacitor. Analog (needle) multimeters may be better suited than digital multimeters as the needle is easier to read for this purpose, rather relying on update speed of the digital meter.
2. To measure the "actual" capacitance you need a proper capacitance meter or you can measure it approximately with an oscilloscope using RC time constant theory.
3. You can also use an oscilloscope with a test circuit in XY mode to also perform comparative characteristics between capacitors.
4. An Electrometer or Capacitor analyzer can be used to measure leakage.
5. However an ESR (equivalent series resistance) meter is a far more useful tool in confirming a capacitors health. Even if it appears good and measures the correct capacity an ESR meter should show if the capacitor has moved way out of spec. Some ESR meters show capacitance and ESR and some are multifunction meters and perform transistor and inductor tests.
6. Kelvin leads (4 wire) especially for trying in circuit testing, allow you to take measurements that reduce the effect of the leads resistance. A more accurate measurement for anything resistance related.
7. But in circuit testing can show inaccurate results due to other resistances in parallel. If you are using an ESR meter or equipment that tests above a certain voltage you can also inadvertently turn on transistor junctions and be testing other components. Therefore sometimes desoldering one side of the capacitor may be required for accurate measuring.
8. Caps generally should be discharged first before testing by short circuiting them (touching their pins with an insulated wire) or using a resistor in addition to this to slowly drain them. Charged caps can damage your equipment, larger caps can be lethal and must be checked and safely discharged prior and during working on the equipment.
The forum is full of threads on ESR meters.