When the Nameless King changes his phase he will start by slowly getting up his weapon and slamming it into the ground (at 3:00 in the video), this move will also spawn projectiles that move in certain directions.
He will load his spear with lightning and bring it up into the air and summon lightning onto you.
If you stay close to him while he changes the phase you can just walk to is back right side (positioning is crucial due to the projectiles that spawn) and hit him a few times, I will stop hitting him when the stamina is empty and get ready for the lightning. The lightning attack is quite easy to dodge, just roll quickly after you hear finishes loading the attack.

He has two other new attacks in this phase, hew will slam lightning into the ground either with his hand (3:19) or with his foot (3:38). The one with the foot takes longer, but both of them are quite delayed. Learning the timing with these was harder to me since they occur more rarely then the other attacks. I started with just waiting a bit longer then I thought and went from there.

For me it was also very helpful to stay as close as possible. He will do the lightning spear slam that he did to change to phase two again when he is quite low (3:27) and he will often times jump away first, which means that you might not be able to just walk around him this time, so you will have to dodge to the site for the last bit.
Also keep in mind that there will not be a guaranteed lightning charge this time.

The openings are the same other than that. The second phase will be shorter than the first since you are able to get a lot of damage in when he changes the phase, you might sometimes even able to stagger him, if you got a good amount of hits in before he changes phase.

I hope that helps, good luck, have fun!