If it's totally dead there's not very much you can investigate except to check that the a/c adapter is working by plugging it in to a different but compatible laptop, or testing the a/c adapter's input connector with a voltmeter.
If the a/c adapter has a power lamp on it, is it lit up when you plug it in to the mains socket? If it isn't, check the plug-fuse. If new fuse makes no difference, still no power lamp lit, you need a new a/c adapter.
You could also try running your laptop from the a/c adapter with the battery removed once you've established that the adapter is working okay.
If nothing above makes any difference, you will have to let a laptop repair shop take a look at it, but generally they are not too keen to do anything...