I now have a total of 5 (yep, five -- four working and one not) Asus ROG G751JT laptops in the family, and the first one we bought for my son started having battery problems after a little under a year. I did some troubleshooting with the help of Asus support and determined it was a problem with the charging circuit on the motherboard, so I RMA'd it to Asus under warranty (which still cost a fortune). They did fix it, but then just a few months later, a cell in the battery failed (no, the b@astards didn't replace the battery when they were in there replacing the motherboard). So I bought a 3rd-party battery on Ebay and replaced the failing one. The battery would charge up fine, but when I tried to run the laptop off the battery, it would abruptly shut down and would only stay on while plugged into AC power. Given that the battery charged up to 100% fine, I assumed it was another problem with the power circuit on the motherboard, but it was already beyond their 90-day warranty on the original warranty repair, plus I didn't want to spend another $200 shipping it back to Asus. So I just shelved it and bought another one (easily moving the hard drive over) and got the poor boy back up and running. When my other son's ROG screen cracked (closed it without realizing something was on the keyboard), I cannibalized the shelved ROG to replace that screen.
So, I recently took our "parts" ROG off the shelf, replaced the LCD and decided to try another battery (3rd-party but from a different source), and this busted up ROG is back to running like new, runs off the battery just fine, screen looks beautiful, etc. The moral to this story is if after replacing the battery your laptop still doesn't work (at least not properly), then it could still be a battery issue and you should probably try another battery. The 3rd-party batteries have questionable reliability, but if you get one that works after swapping it out, it will most likely stay working fine. I was going to use this ROG (working battery or not) as a security camera DVR, but now that I have it working perfectly again, I'm just going to keep it around as a backup for when/if any of the other ROGs fail (it's so easy to just swap out the hard drives).
Replacing the battery isn't that difficult. You do not have to remove the motherboard. If you have a very thin shaft phillips head screw driver, you can even remove and reinstall the two screws on the battery without removing the screen/hinges/etc. (which is great because reinstalling those wifi wires on the motherboard is a tedious pain!). Once you have the two screws out of the battery, just use a flathead screwdriver to carefully flex the bottom/back of the casing out and then use another flathead to pry the battery out. You're just looking to get the battery out over those four locking tabs. Just be careful how much force you use so you don't crack or otherwise damage the laptop casing.
The LCD is even easier to replace than the battery. On YouTube there is a really good tutorial on the whole process for a G750 series -- don't follow that! It's very different from the G751 series laptop, and on the G751 you literally only have to remove ~8 screws (4 on the bottom edge of the screen casing and then 4 more on the inside that holds the LCD into the frame). There's one main ribbon cable and a very small ribbon cable for the camera.
The replacement LCD was $100 and the battery was $55. That's cheaper than shipping to Asus for a warranty repair.