Your first step in this process is to talk with your Human Resources or Personnel department. They can help you discover what the policies of your organization are for promotion and the specific qualifications for various grades.
As you write your letter, there are several things that can influence how persuasive you are. The first is that the letter should be well written and well organized; the quality of your writing exemplifies your written communication skills, which are a priority for many companies.
Next, you should focus not on your wish for more money, but on your current and potential future contributions to the organization. In other words, saying that you want more money so that you can live a better lifestyle won't be convincing (we all would like more money!), but instead you should focus on
- Measurable successes in your job
- Evidence that you have been performing work at the higher grade level
- Specific ways in which you meet the criteria the organization has for that grade level
Finally, you should conclude by asking what you can do to improve your qualifications for more senior positions.