Basically, you are applying for a (more senior) position with the company but doing it internally and without an application form. You need to write that you are applying for such and such a position as you now feel that you can take on the extra responsibility - describe how well you have undertaken your current job (successes, achievements, ability to get on with the rest of the team, what you've learnt etc) and how your known strengths can benefit your employer. Mention your willingness to learn and develop and bring further benefit to the company. As you already know the company, its products, clients, staff etc there is the advantage in promoting from within, plus they know you and your track record. Also mention how you would deal with you not being employed in your present position - you could offer to train or mentor your replacement for a short period of time to smooth the handover procedure.
Good luck !