"Melancholic" is a Japanese movie which genre is suspenseful comedy movie.

"Melancholic" is a Japanese movie which is a suspensful comedy. -> Another way to say this, but still a little bit too long. The usual way we describe movie genre is by adding them as adjectives. "Horror movie" "Romantic movie" "Romantic horror movie" etc.

One day, he runs into a woman who was a n ex-classmate at a public bath.

However, during the night, the public bath is used by a gang killer who is hired by a gang in order to killmurder people...

Killer to kill people sounds repetitive, but is not wrong. Changed it just to sound better. Gang killer is also just to make it nicer sounding.

Some parts wasere unreasonable, but it iwasn't a problem because it's a comedy.

"Some" tells us more than one part was unreasonable, so you should have parts be plural.

I thought the main actor well expressesd a man who is intelligent but doesn't have good communication skills.

"Melancholic" is a Japanese movie whichose genre is suspenseful comedy.

The main character is a man who graduated from a distinguished university but is working a part-timedead-end job, spending every day without purpose.

'Part time job' in English doesn't strongly imply a job which is a waste of time, as I think you mean here. Dead-end job is a great word for this scenario, although this could also be a full-time job.

One day, he runs into a woman who was a classmatehe went to school with at a public bath.

There are lots of ways to say this naturally: 'former classmate' 'old classmate'. These sound awkward with the word 'female' though, so I've changed it as above.

However, dDuring the night, the public bath is used by a killer who is hired by a gang in order to kill people..hitman.

'However' isn't natural here because, although this might be surprising, this isn't a counterpoint to what you said previously.

It's not entirely clear how the bath is 'used' so I've left the sentence unchanged. However, you probably need a little more detail.

Some parts was unreasonere unbelievable but ithat isn't a problem because it's a comedy.

Or 'far-fetched'.

The story was intriguing, and the actors were nicegood.

'Nice' makes it sound as though they are kind, caring people. This isn't something you can tell watching a movie. Saying 'good' implies they are good actors.

I thought the main actor well expressesportrayed a man who is intelligent but doesn't have good communication skills well.