Authorized to write a personal application to higher official regarding personal information

If you know the senior officer personally and he is good for you it is always better to go and talk to him about your personal matter. Why write letters to them unnecessarily which may be viewed by others in a different view. It will not be good for both of how.
If you want to write you can write an ordinary letter to his house address. But it should be like a normal letter we write to our friends and relatives. In that, you can ask his advice on the matter.
Trying in an illegal way and trying to use the influence and pressurising the concerned person for posting will lead to disqualification of you for the post. So such things should not be in writing.

Writing a personal letter to higher ups for a favour on personal grounds is against the conduct of employee service. It should not be done as it can be viewed as a measure to ask or affect the judgement of higher ups for securing a favour. In any case it should be avoided. If you know the higher up officer personally then you can talk to him in person if he is ready to entertain you for the said matter. Some personally known people do not talk official matters in person so you must be cautious to that extent.

If an employee has a grievance or complaint then he can write letter to the grievance cell and give a copy to the department. This is acceptable and the grievance cell will look into the matter for necessary consideration as applicable under the grievance procedure.

If you want to have any personnel problem resolved with your boss, you may do so with his permission. It would depend upon the nature of your boss to pay heed to tour personnel issues. If you are fully aware of his temperament, his patience of hearing and sensible in nature, you can go ahead with your issue to be resolved after taking an appointment with him either in the office or at his residence. But you need to apprise him of your issues which you would like to discuss.
Writing an application indicating your issues and get it resolved in connection with the official problem, is acceptable with a duplicate copy to be forwarded to higher up is acceptable but ensure the maintenance of code of conduct of the office. Any violation of such rule may attract to disciplinary action against you.