What's the proper way to add new gravel to existing tank?

This shouldn't be too much of an issue if you are just adding more of the same gravel (or any regular aquarium substrate) to the tank. Rinse it like you normally would, and put a little at a time into the tank and spread it around until you have the depth you want. Then adjust any aquarium ornaments. It should be fine unless you have live plants or fish breeding in caves or something.
 
It shouldn't, but you could mix the gravel together some if you like. Even if you have an undergravel filter that uses your gravel as the main biological filtration it should be fine, as the filter will still pull water past the old gravel. However, in a tank with another type of filtration, your gravel bed is not the main source of bacteria in the tank anyway.
 
AquariaCentral.com