The wiki has a basic strategy for taking out the Hellkite Drake.
In earlier versions of the game, the Drake did not have a regeneration factor, making it more of a marathon, but fairly straight forward to take him out. There were also a couple of exploits* that could be employed to take him out, but have since been patched in later versions.
Now, the main consideration to account for is his Regeneration. As @bgalin mentioned in his answer, once his health drops below a certain point, he covers his head and begins to regenerate his health, making it nigh impossible to kill him using a ranged method, depending on your stats. This only becomes more potent with each NG+.
Using a melee approach nullifies this, as he is constantly aggro'd, and never returns to his perch (so long as you keep in his focus, and not retreat). The only considerations here, are Fire-resistance; like using "Flash Sweat" pyromancy, and high Fire-resistant gear, and Lightning damage. You can imbue a weapon, or use the Gold-Pine Resin as a quick alternative.
*There used to be a bug, where if you climbed up the tower behind the Hellkite, and shot him in the head, he would aggro, and jump down onto the bridge (which wasn't actually rendered), at which point it would plunge to it's death.