How can I write this letter for my story?

Only you know where your story is going. Ask yourself what was the brother's motivation. Was it revenge (perhaps he was jealous because the younger brother was the favored one)? Is it a guilt-ridden confession? Perhaps it's an explanation if they were close and he intends to disappear. Is it an apology? Is it a suicide note? Write down as many scenarios as you can. Leave the story for a couple days while you mull over your choice and the implications on the rest of the story.