Earlier today, I turned on my laptop on.

Everything was okay until I plugged it in to charginge.

It’On its screen appeared something strange.

I don’t know what ist its.

But It looks likes a lot of white lines running along the screen.

It covers all of the laptop’s screen.

I will replace newthe laptop’s screen with a new one next week.

Everything was fine, until I plugged int in to charginge.

I hope this is what you were trying to say? You didn't have any adjective in there, so I put "everything was fine".
Instead of saying "I plugged it in charging", you say either "I plugged it in," "I plugged it in to charge," or "I started charging it."

It’sOn the screen appeared, something strange appeared.

You use the preposition "on" when talking about screens/phones/computers etc ... anything *on* the screen is something you can see.

I don’t know what is it.t was,

Bbut Iit looked likes a lot of white lines running along the screen.

You can combine these two sentences :)

It covers all laptop’sed the entire screen.

You used the possessive ('s) right, but you wouldn't use it here. You can either just say "laptop screen" or "screen". It sounds more natural to say screen.

I can’t see anything, so I can’t do anything on my laptop.

I tried restarting my laptop.,

Bbut this wayrestarting didn’'t workdo anything to fix the problem.

Sounds better like this.

I will replace new laptop’sthe screen next week.

When you say replace, you don't need to say new, because replacing something already means you will get something new.