I was actually doing some research on body knockouts last year, and I can give you some of the information I've dug up. It's not as detailed as what you're looking for ("best methods"), but it is some useful information nonetheless.
Liver, Location, Access
The liver sits under the right side of the ribcage. It's mostly, but not entirely protected by the ribs. Why mostly? Well, there's a host of reasons the liver can end up with it's lower half below the ribcage, including:
1) Depression downward when the diaphragm is depressed when people have full lungs
2) Visceral drift - the viscera in the abdominal sack aren't tightly locked into place, they are held mostly by the network of blood vessels and fascial connective tissue. The stomach is the most common organ to migrate around from standard, but other abdominal organs can vary position as well.
3) Anatomical variation - some people have larger, or smaller livers, and sometimes the variant can have the liver covering from the front to 2/3rds around towards the spine. Likewise, taller/shorter and rib coverage.
This medical article talks about the liver typically being accessible 2-3 CM below the rib margin, my experience in bodywork typically finds most people have about 1-2 inches available.
(Fun sidenote: you can do test this on your friends and training partners by tapping on person's abdomen, the liver is denser than the intestines and gives a different sound. Doctors sometimes also do this when they're trying to isolate your lower right lobe of your lung to place the stethoscope above the liver.)
Delivering the Strike
I have no clue what's the best way, but I see several styles and places give a thought to how to get in there and deliver it. It shows up a lot in boxing, for pretty obvious reasons, and it follows from a basic jab or 1-2 followed by a digging hook to the ribs. Here's another video with a few combos to get into a liver shot.
EDIT: I've found the Muay Thai liver shot video. They talk with a doctor about the anatomy, then go into some of the ways to get hits in both with hands and kicks.
As far as I can tell, pretty much every style that puts thought into doing it mostly looks for how to open up the opponent to drop a power strike in. Whether one is better than the other, is something I'd want to have some practice with or at least talk to someone who's gotten it to work in different situations.