How can I get a koseki record of my father?

I want to know his whereabouts now. But the only information I know about him is his name, birthday and his address and number 16 years ago. I have his pictures together with my mom and some EMS. Also, I am a child out of marriage but he knows about my existence since he was supporting me financially but it stoped when I was 3.

I dont think you can just ask for it. Only the person who is mentioned on the koseki can get his own information.

It would be strange if everyone can get the information about someone. I cannot also not just go to the city hall and ask my wifes koseki

if you get a koseki tohon, it does not have information about his current address.
I think you need to hire a lawyer.

I cannot also not just go to the city hall and ask my wifes koseki

Anybody listed on the koseki, including foreign spouses, can get a copy.

A few things to consider:
- can you find out from your mother if your father officially recognized you, meaning, put you on his "koseki" as his, though out of wedlock?
- if yes, and you are registered as his child on his "Koseki," then there might be a chance. But it might not be at the same address as the one you have if his (the old one), so you would need to inquire with that city hall, or hire a personal investigator or a lawyer to do that on your behalf.
-if he didn't officially recognize you but just provided financial assistance, and he has moved away from the old address, that could be the end of the clue.

Still I am not sure what you are after: are you trying to find out where he is now, or are you trying to obtain an official copy of his koseki?