The Yeti King can be relatively easy to deal with as he isn't actively hostile to The Spelunker, but there are two factors worth considering. First, he always spawns with a horde of smaller Yeti, and secondly, his roar can cause avalanches of ice blocks to fall, crushing anything they land on.
The best way to kill the Yeti King is by catching him in a bomb explosion, which will kill him instantly. If you're lucky enough to have a UFO spawn near him, you might be able to throw a projectile at it and have it crash down on him and explode. You can also use a Shotgun, should you be lucky enough to have one, but be warned that he'll take more than one blast to be killed. However, a well-timed shot can sometimes push him into his own avalanche, insta-crushing him.
Additionally, bouncing on his head repeatedly is an excellent tactic as he moves quite slowly and has no way to shake you off, assuming you can move through the hordes of Yeti to get close enough to him.
As long as you can chain jump reliably, he should die with little trouble. Thrown objects are also effective, especially as they can cause a chain reaction with the hordes of Yeti around him. In fact, stunning a Yeti and using it as a projectile is also very effective. His width and slow movement means that most thrown objects tend to hit him multiple times as they move through his hitbox.
Sometimes when he roars, the ice that falls will crush the Yeti hordes around him.