You'll need as many arrows as you can get your hands on, and also the upgrade from Apollo to shoot arrows rapidly. You'll also need plenty of magic orbs, but not necessarily as much as the arrows. I beat this boss with about 200 arrows and 70 orbs, just about running out of each.

As you probably have already seen, the Lich King has 4 attacks, and like other bosses when he's about halfway dead his attacks speed up / get more intense

Lich attacks:

1. Center platform, lightning bolts shoot across the screen
2. Top left platform, fireballs rain down
3. Top right platform, skulls come from left and right
4. Bottom left platform, spiky shield dudes come from right

Here's how you deal with each one:

1. Pretty simple, you need Selene to transport through the lightning bolt as it comes across the screen. I found it best to stand under the center platform, transport through, then get in as many fireball attacks as you can.

When it speeds up focus more on not getting hit than getting attacks in, since the lightning bolt can do a lot of damage to her.

2. This one gave me the most trouble until I figured it out. Selene can vaporize a fireball by teleporting right under it, but the timing is really hard. So, have Pampas lay 3 landmines on the center platform, and then stand underneath it as the fireballs get destroyed by each landmine.

Repeat as needed, and this should also work for when things speed up. You have roughly enough time to get in a fireball attack with Selene in between laying landmines, but when it sped up I found this too risky.

3. Fairly simple, use Pampas' sword to kill the skulls keeping in mind they'll vary in coming from the left and right of the screen. You can also get in some sword attacks in between or try to quickly switch to Selene and get a fireball in.

When it speeds up, I couldn't quite handle things with my sword so I used arrows on the skulls instead.

4. Only Selene can damage these guys, and use your lightning attack since fireballs don't work on them (with the exception in the next paragraph). Before things speed up, you will have ample time to damage the lich with either fireballs or Pampas' sword.

After speeding up, 3 of them will come one after the other and Selene does not have enough time to kill them with the lightning attack. So use her transform spell (down and B), then hit them with lightning or a fireball to turn them into sheep. The transform spell takes a second or two to load so be as quick as you can. You can try to sneak in some attacks on the lich before this, but be careful since if you're too late the spiky dudes will rapidly drain her health.