how to repair the iPhone 5 which buy from New York but now I am in HK

Probably not because the iPhone warranty is NOT international, but is country specific, so if you purchased it in the US, then it will have to go back to the US for repair.

You can try and take it into a store in Hong Kong and see if they will help you, but if it is a US phone, then they have every right to refuse due to warranty cover and they also have different models if your phone needs replacing and they won't be able to give you the same model if the phone came to from the US and this is why warranty cover for the iPhone is country specific and NOT international.

You can try, but you might be out of luck and if so, then there is nothing you can do except send the phone to any relatives that you have living in the US and see if they will take it to an Apple store in the US to get it repaired for you. Otherwise you'll have to get it repaired on your next trip to the US.

This is exactly why you should have thought very long and hard before purchasing a US iPhone and having exactly this situation where you can get nothing done about repairing the phone due to country specific warranty.

The only other thing that you can do of course, is to buy a new iPhone from Hong Kong and then you will have no problem with warranty cover.