Just like the "Enkindle"
achievement, I suppose skipping this achievement is definitely possible, but again, good luck. Estus Flasks are health potions, so if you can somehow beat the game without ever acquiring them, you're inhuman.
It's really not hard to miss this one if you are a normal human. The asylum only has one direction to go all the way through, so if you're lost I really can't help you short of playing for you. After lighting the bonfire, the only way to go is through the giant double doors. An Asylum Demon
will jump down from above, and you can either fight him or skip him this time. You're welcome to fight him, and if you manage to kill him, you get the Demon's Great Hammer
, but if you skip him for the first encounter you can't get that weapon again.
After running past him, you can enter a room to pick up a shield, and another for your weapon depending on the class you started with. Then you'll come into another section on the second floor. Once again, only one way to go so you can't really miss it: Up those stairs. Be careful though, a boulder is activated and will do a good amount of damage if you don't dodge it. HERE IS WHERE THE ACHIEVEMENT COMES IN
That boulder knocks down a wall, and entering that little area reveals a dying Elite Knight. Talk to him, and he will give you his Estus Flask, as well as that sweet, sweet Gamerscore. Here's a map of the asylum if you can't figure it out.
Praise The Sun!