The Humayun Nagar Post Office is the FPO for Hyderabad, Foreign Post Office, you don’t need to go anywhere else to send.
If it’s ordinary post, you don’t even need to explain anything, just go in from the side entrance (not the business center) and say you have a parcel for ordinary post and they’ll tell you which basket to drop it into.
But for this you’ll need to weigh at home and buy stamps from the General Post Office in Abids or online.
Anything else (speedpost, EMS) you’ll need to open the parcel and explain, but there are no screenings or inspections for ordinary post.
But you will need a customs declaration form pasted on the outside:
I’ve shipped game consoles to France, processors to Yemen and stickers to Japan, all through ordinary post and a bunch of stamps.