How do I kill Kaos?

You can NOT actually KILL Kaos. The best you can do is defeat him in the final battle in his Castle. To get to that, you must complete all the lower chapters, finally completing the Lair of Kaos. You don't have to get all the check marks in "objectives". You don't HAVE TO collect all the soul gems, tresure chests, legendary treasures and scroll, to get past Lair of Kaos. However to COMPLETE the entire game, you have to collect EVERYTHING, KILL EVERYTHING, COMPLETE ALL THE HEROICE CHALLENGES, AND DEFEAT KAOS. That is the only way to get 100% on game complete. Anything short of that and your completion percentage will be below 100%.

Once you have completed Lair of Kaos, the game will take you to Kaos's castle. There are not treasures or scrolls or gems to collect. You just have to FIGHT All of Kaos' horde of Minions. That fight takes a LONG LONG time. I own 30 of the skylanders, plus all the legendary. I even own Dark Spyro, which means I have 3 spyro's in my arsenal. ALL of them have been leveled up to 10 and all but 2 have purchased ALL the upgrades from persephony. The final upgrade can't be purchased until you collect the soul gem in lair of kaos for Double Trouble.

Even with that HUGE arsenal of good guys, it took me nearly an hour to fight and defeat the wave after wave of minions Kaos sent against me. Repeated Sharkwave of Doom attacks, laser attacks, those green balls, and of course wave after wave of sets of 3 minions at once. SO many that I lost count in about the first ten minutes of the fight.

I think he used his entire arsenal, and I just managed to dodge enough and get in enough of my own hits to finally wear that Hydra down to zero energy.

I tried shooting Kaos and his Hydra, with everything in my arsenal. I got a lot of direct hits, which I believe weakened the hydra a little.

The main thing is: to defeat Kaos, you must somehow manage to survive all the attacks. EVENTUALLY the hydra will run out of energy and you will defeat win. The trick is to have as many skylanders as necessary to KEEP replacing your exhausted ones, so you can keep fighting.

I am going to try again on another new game and use all 8 of the little add-on gizmos (one at a time) that came with the adventure packs to see if that helps reduce the body count. I really don't think it is possible to defeat Kaos with only ONE Skylander, but I could be wrong. Dodging all the "incoming" was virtually impossible.

I also plan to try one of the water skylanders that regains his health in the water. (sharkbath of doom has lots of water)

I suspect this design (requiring LOTS of Skylanders) on the part of Activision was intentional. If you don't have as many Skylanders as needed to defeat Kaos, you will but more and try again.