Charge thru a resistor to the working voltage. Choose a resistor so RC (where R is the resistance, C is the capacitance, and RC is the time constant) is workably large. The final voltage should equal the applied voltage - IR, where I is the leakage current. The rate of charge will give you C ( if I is large you will need to correct for that ) This ignores the burden of the meter which is probably above 1 meg and for a supply cap probably does not matter.
It IS possible to store high voltage energy, and probably possible to built a store that would not be destroyed by the first lightning strike. BUT even the biggest cap easily available will store less energy than a 2500 mAh NimH AA cell. Many such capacitors would allow significant energy to be stored "at a strike" but the store cost would be large.
It would be possible to construct capacitors capable of say 10's of kV rating which were protected [tm} by spark gaps so that the gap broke down well below capacitor destruction voltage. You would then potentially (pun noted) have a problem with back conduction of the capacitor via the ionisation path, but could very probably build a system that used a spark gap from input to capacitor to initialise charging and then another spark gap from input to ground such that it reduced the available potential of the charging spark gap and extinguished the arc. While this sounds rather "Heath Robinson" it is similar to the sort of things that are really done at extra high potentials.
As an example of spark gaps being used as switches, a "Marx Generator" uses arc breakdown to cause spark gaps to act as switches and allow construction of a voltage multiplier witj DC input. Good trick if you can do it. You can ! :-)
See "Marx generators below.
Capacitors:
You can make DIY multi kV capacitors aka "Leyden Jars - with glass jars or bottles - or even plastic "pop" bottles. From here
Here's a 19 nF 10 kV cap made with foil and mylar shopping bags
You can buy capacitors in the nF range rated at 10's of kV.
Many offered here - Alibaba
Example from above - 40 kV, 10,000 pF disk ceramic.
At 40 kV 10 nF will store 8 Joule.
About the same as a 0.64F cap at 5 Volt.
Many very high voltage caps here - very worth looking at
1.3 uF at 100 kV !!!!!!!!!! :-)
6500 Joule.
Less than the energy in a 2500 mAh AA NimH cell (= about 10,000 J)
Marx Generators - as examples of spark gap switching.
This video of a 180 kV output Marx Generator operating is worthwhile just for the sound :-).
I am NOT suggesting that a Marx generator is what you need to meet your suggested requirement - but offering it as an example of how switching may be achieved by quite unusual but eminently practical means.
Wikipedia - Marx Generator
The following is far from the most impressive spark-gap-switch unit image available - I chose it because it gives a good view of the spark gap switches working (small arcs along the centre line) plus the resultant overall multiplied voltage arc (you should be able to find that without my guidance :-) ). To make this work you only need a voltage able to break down the small spark gaps.
Worthwhile
LOPT in, 200 kV out](http://c4r0.elektroda.eu/_hv/index.php?page=hv/marx) Polish language.
Project with image
Very good project description.
"Quick & Dirty" Marx generator
Karl would have been proud!
Best Answer
One big issues with consumer electronics is that they use the cheapest components possible. Electrolytic capacitors in particular are notoriously failure prone.
When replacing failed capacitors it generally a good idea to replace them with ones of equal (or close) value but with a higher voltage rating. For example if the manufacturer used a 6 V capacitor on the 5 V line, you should consider using one rated for 10 or 15 V.
You probably did not get bad capacitors, assuming a reputable source, but if the replacements had the same ratings as the original, and the originals were marginal (and thus failed), the replacements were probably marginal as well.
I should probably add that heat is not your friend. How is the cooling in that part of the board? If the air vents are clogged with dust etc. that could cause them to fail as well.
If you would like another opinion see this EEV Blog forum. The conscious seems to be that you can't go wrong with Panasonic or Vishay capacitors but that many of the lesser brands will be nothing but problems. (Of course there are other good brands as well).