To get this you need to have at least beaten the Folding Screen Monkeys boss in order to unlock the Puppeteer Ninjitsu ability. From here travel to the Senpou Temple, Mt. Kongo sculptors idol. Exit the building and head north east down the cliff, go left past lone monk on the broken bridge and drop down, you should see a lone rat enemy near a mechanism. Stealth kill him then press
to activate the Puppeteer Ninjitsu ability (you need to have this equipped), he will then reel in the kite.
Progress through the area and then through the next building. Upon exiting you should see a very large tree in front of you, you need to grapple to the top of this. Once at the top you will see the kite used by the rat, you can grapple this and traverse the canyon. Once across jump off the ledge to the right and use the grapple multiple times until you reach a cave, this contains the Sunken Valley Cavern sculptors idol.
Travel through the cave until you reach the Sunken Valley Passage, you will see what looks like a plank overhanging the cliff. Walk to the end of this and you should see the Great Serpent below you, simply jump on and perform the deathblows when prompted and the serpent is no more. Enjoy your achievement!
Any issues please let me know!
*EDIT* You need to have encountered the Great Serpent in both the Ashina Castle Outskirts as well as the Gun Fort before he'll appear at his killable location.