I'll try not to alter too much of the sense of your paragraph. Here are my corrections in bold with a few notes.

Good morning, everybody! make sure to separate your greeting with a comma (,) followed by who you are addressing. In this case, "everybody".

Now, it's 6:23 AM. There's no need to use o'clock in this case. In normal conversation, one would only say "o'clock" if it was 6:00 (Six o'clock). If it was 6:23, you would say "It is six twenty-three."

I am going to tell you two things. Going is typed "gonna" if you choose to use slang.

The first thing is that I decided to write at least one composition in italk to improve my english. The second thing is about my dream. In the dream, I dreamt about boy who I liked in my primary school - He is handsome. Unfortunately, when we graduated from primary school, we fell out of touch. He left a deep impression on me. For a long time, I missed him. Recently, I keep dreaming about him. In my dream, he still looks like he did in primary school. It seems like he never changed. Though, one time, I looked at the photo taken for the primary classmate party and I noticed that he changed dramatically. He is no longer handsome. He is fat which frustrates me. However, I can't help dreaming of him. In the dream I like him, but when I wake up I can't describe my feelings. Yesterday, I dreamt of him once again. I don't know why. I want to share this story with all of you. I want somebody to listen.