Well, I'll try to correct though my English is not very good.

So:

I stood alone, silent, at the window.
It was pouring rain on the wall, everything was as always, - The phrase "Лил дождь стеной" means that the rain was so dense, as a wall.
But then, I heard steps from behind,
They altogether came closer to me. - No, the words "всё ближе [и ближе]" here translate as "nearer and nearer".
"Who are you, and how did you get in here?"-man
A cold fear pierced my voice,
but he calmly lifted (his) eyes - means he raised his glance, looked up, raised his eyes.
"I am the angel of death, it's time for us to go." -Angel of Death

In the void, he fell, with his hands still clinging to life. - To the empty space.
"What do you say to him? Look at his actions" - Some other angel? (don't understand what another angel are you talking about)- A better translation is "What can you say (or "what is there to say") to him. Look behind yourself at your actions".
"In the void, taking for himself the souls of those who have so often sinned, - To the void (empty space)
and to ask for forgiveness is already too late, you yourself having decided everything." - Is this even being said or is this back to past-tense storytelling mode? - It's a voice-over, a story-teller phrase.

I didn't manage to do anything in life here,
No family, no one close, no friends, - "близкий человек" is someone loved, not necessarily in a sexual sense, but some who is very dear to you.
I forgot about everything, lived for myself, - No, the phrase says "Я позабыл про всех", not "Я позабыл про всё". The word "всех" means people, not things. So the better translation is "I forgot about everyone", or even "I never cared for anyone".
I even lost my love. - The word "love" here means "the loved one".
"I do not care, you are just a man,"
and removing his coat, he spread (his) wings. - It's better to say "cloak", not "coat".
"you did not try to change anything,
and the time comes for Him to judge you." - Angel of Death

Returning again, to where I was happy, - To return (he asked)
He knelt to say - He was saying kneeling
"Fix what was done." - Angel of Death - No, these are the words of the man. He is kneeling and asking the angel to let him return to where he was happy and fix what he has done.
I understood it all, and did not appreciate it before.
"(She) may have the strength to forgive me.
At least give me a chance to speak with her!" - man
He, begging, looked down
and arose from his knees, and the room was empty.