"The Idiot" was histhe first novel of his that I read.

He writes a letter about how he feels like he's being sentenced to death inat such a young age.

The first in the list of my favorite authors list will be Fyodor Dostoevsky.

Many people heard of or read his novels of his, such as "Crime and Punishment", "The Brothers Karamazov", "The Idiot" and others.

"The Idiot" was his first novel of his I read.

Then I re-read it several times because of the novel's characters.

I think the most memorable part for me is a chapter about Ippolit, a young man who is going to die of the phthisis.

He writes a letter about how he feels being sentenced to death inat such a young age.

In "The Brothers Karamazov," I like a chapter about a monk the most.

It tells about us a hussar who becomes a monk after he goes on a duel, and realizes how wrong it is beingto be forced to kill a man.

"Tell" in the sense more of "inform" than "recount a story" needs an object.

Then he meets a strange man, who is admired byfor his action,s. He becomes his friend and reveals hisa terrible secret.

In the end, the man tells truth about hisa crime in the past to the citizens of ahis town, and tragically dies not being able to carry the aftermathunable to suffer the consequences of his deed.

"Unable" sounds better than "not being able." I also find "carry/bear the consequences" too vague. I looked it up since i couldn't remember that subplot, and it sounds like the issue was not that he couldn't come to terms with it (which is what your sentence suggests) but that no one believed him.

And, lastly, there are several shorter stories, which are rather good.

"Stories" without qualificiation in the context of literature suggests "short stories," but that doesn't fit "Notes from Underground," which is considered a novella.

The first is an inspiring story, about a man who wants to commit a suicide, but seehas a dream that stops him from doing that, and directs him onto the good path .

The second one shows us athe misery of a human-being.

It tells about a man who understands the evil nature of his actions, but still makinges them, and rationalizeing them with his own dirty little logic.

The first in my favorithe authors list willould be Fyodor Dostoevsky.

Many people have heard of or read his novels, such as "Crime and Punishment"," "The Brother's Karamazov"," "The Idiot," and others.

Adding "such as" makes the sentence/the transition into the list sound smoother.
If you are writing in American English, you put the commas inside of the quotations. I'm assuming you are writing in American English, since earlier you wrote "favorite" instead of "favourite" (which would be British English) :)
But if not, then you were originally correct.

"The Idiot" was his first novel that I read.

You can also say:
"The Idiot" was the first novel of his that I read.

Then, I re-read it several times because of the novel's characters.

"Then" is a transition word here, so you need a comma

I think the most memorable part for me is a chapter about Ippolit, a young man who is going to die of the phthiuberculosis.

I actually had to google to see what "phthisis" meant! It is an extremely old word for what we call now "tuberculosis."

He writes a letter about how he feels being sentenced to death inat such a young age.

In "The Brother's Karamazov," I like amy favorite chapter is about a monk the most.

In "The Brother's Karamazov," is a transition, so it needs a comma.

Sounds better to say "my favorite chapter"

It tellis about a hussar who becomes a monk after he goefights oin a duel, and realizes how wrong it is being forced to kill a man.

Then he, the monk meets a strange man, who is admired by his action,s. They becomes his friends and he reveals his terrible secret.

In the end, the man tells truth about hisa crime inhe committed in the past to the citizens of a town, and as a result tragically dies after not being able to carry the aftermathhandle the consequences of his deed.

"aftermath" is a good word, but I changed it to "consequences" since I added "after" a bit before. I added "after" because it sounds better to put a timestamp in there.

And, lastly, there are several stories, which that are rather good.

I personally like "The Dream of a Ridiculous Man" and "Notes from Underground"."

The first is an inspiring story, about a man who wants to commit a suicide, but sees a dream that stops him from doing that, and directs him on the good path .

The second one shows us athe misery of a human-being.

It tellis about a man who understands the evil nature of his actions, but still making themnevertheless continues to act the same, and rationalize them with his own dirty, little logic.

I would use the word "tells" when someone is talking, not what a book/article/show etc. says