The reason that the grievance process sounds like "a legal process for solving problems" is that it is indeed a pseudo-legal process for solving problems. In particular, it is used when the instructor has done something that violates university policy - unfair grading is a frequent complaint.
These grievances typically follow a detailed policy to try to ensure that student complaints are treated fairly. The first step is often to formally notify the instructor in writing, and that seems to be the purpose of the letter.
In the letter, you should lay out the facts as you see them, and you should include the resolution you would like to see. Try to write the letter in a businesslike way - you want to advocate for yourself, but keep to the facts and try not to say things that you cannot justify later.
If the instructor and department chair cannot or do not resolve the issue, it will move up to the next level, which is typically review by a dean. If the dean cannot resolve it, it goes to the next level, which may involve a hearing of some sort. In my experience many complaints are resolved by the department chair or dean, particularly when the situation is clear cut (e.g. a professor didn't follow the grading policy from the course syllabus).
I cannot say what remedies will be available at your school - this will vary by school. There are certainly colleges where the higher-level administration could allow you to take the course again, or drop the first course from your transcript. The instructor will not be able to do those things, but you have to work through the process one layer at a time in any case.