No adequate system in staffing
I work for a technical institution in governmental sector. I have 10 years of work experience and I regularly perform job functions that go beyond the scope of my original job description, but I believe that my institution has no adequate system in staffing and upgrading personnel in place.
In addition, my direct supervisor (boss) has been transferred to our office recently. Unfortunately, he was appointed a head of department. I believe that I am more competent than him in terms of technical skills, experience, communication skills, Arabic/English language skills, as well as leadership skills.
It may be time to propose a change in position to better reflect my professional competency. I am looking forward to a higher position. Could you please advise me on this problem?
Waiting for your kind response

Why do you think your “institution has no adequate system in staffing and upgrading personnel in place.”? I feel sure there must be a formal system in place in an organisation in the government sector. Do you have an employment handbook or website available where you can check what policies are in place? If so, consult them before you say anything. If not, ask your HR department for details. Once you understand the policies, you will probably be able to see how best to approach the question of your promotion.
There are probably zone or grades in place, with entry requirements – do you meet all of them? If not, you need to correct that. The list of competencies you describe may not be what is required for promotion. Find out what is required, or preferred, to move up to the next grade, and set out to achieve it if it is missing.
Once you are satisfied on that front, you probably have to wait for a vacancy to arise before a promotion can be sought. But it is no bad thing to make your ambitions known to your manager. Discuss this at any appraisal or performance management meeting you may have.