Yesterday , I spoke with my friend over Skype but we could not hear each other because the internet connection was so bad.

But my My friend thought that my earphones did not work despite itthough they are new.

or,

At first my friend thought that my earphones did not work though they are new.

Despite this problem, we spoke by writting down our conversation or during our one hour session.

or,

Despite this problem, we spoke by writing down our conversation throughout our one hour session.

I have to fix the problem of my internet.

or, more commonly said in the US,

I have to fix my internet problem.

Before I had onI had a good connection before but it became very expensive.

Yesterday , I spoke with my friend over Skype, but we could not hear each other because the internet was so bad.

ButHowever, my friend thought that my earphones did not work despite ithem being new.

Substituted "however" to prevent repeating repeating "but"
Earphones are plural, thus "them"; also "despite" requires you use another verb or gerund: "despite them being new" or "despite that they were new" (I prefer the first option though)

DespiteRegardless of this problem, we spokewere able to speak by writting down our conversation or (missing word here?) during our one hour session.

Substituted "regardless of" to prevent repeating repeating "despite"
"Regardless of this problem, we spoke" is grammatical, but I feel like the addition of "were able to speak" improves the logical flow of the argument: you weren't able to communicate before, now you are

I have to fix the problem ofwith my internet.

Before I had one butI had a similar issue in the past and it became very expensive.

That's why I amwill be forced to find another solution soon.