Unfortunately there is no way of finding out if there is a problem with the charger with a multimeter etc that I know of.
The best way and most definate way is to do as I mentioned, try yours on a friends, or a freinds on yours. One other thing to bear in mind is that either lead can be the problem, what i mean by that is that most charger leads have a 3 pin\2pin plug end connected in to the AC\DC transformer end. So you may just need to replace one end if indeed this is your problem. Obviousley another thing to do is check the fuse has not blown.
There is also no way you can check the motherboard. All you can do if you are capable is to eliminate the charger, internal power jack (by replacing), replacing the CPU. The latter 2 comprise of stripping down the laptop.
Your laptop is basically the LCD and the motherboard, thats what costs the bulk of the money. Its a fine line between if you want to pay say £200 for a motherboard (maybe more), or buy another laptop. If just using laptop for surfing and word processing, then you can get brand new one to suite your needs for around 300 English pounds. It really depends on individual needs\requirements etc.
You really need to get the charger checked out first and decide on the next step after that, this is something that is easy, you can do yourself if you can find a friend who uses a 20v charger, and dare I say may be the most likely cause without getting your hopes up.
Hope that helps in some way sam1111,
Let me know if I can help further (or any BC member),