Will steaming fish cause salt to come out or cause fish to fry in its own fat? If you brine some fish and then boil it in water (make a soup) ,the salt and other nutrients leach right out thus defeating the purpose of the brine.  I will soon buy a steamer, however I just wanted to know if you brine fish and then steam it, is it likely the salt will leach out, move onto the surface or will it stay in the fish?

Also if you grill food, the fat comes out and falls of it, however I have noticed the meat is lightly frying in its own fat as it moves along the meat.  Would the same thing happen with steaming, i.e. fat will come and drip out or lightly fry in its own oil as is the case with grilling?
Salt moves due to differences in concentration, from regions of high concentration to regions of low concentration. So, when you boil brined fish, you surround the salty flesh with low salt concentration liquid, and thus the salt moves out to equalize the concentrations.

Steaming the fish will not result in as much movement of salt, because steaming applies less total liquid to the surface of the fish. Steaming works because the very hot steam condenses onto the cooler food, and by condensing it liberates heat (which transfers to the food). However, you still will have some movement of salt because the condensed steam on the surface of the fish remains at a lower salt concentration.

Also, steaming is much gentler on the fish, since the meat is not being constantly agitated by boiling water. With less agitation comes less leaching.

Grilling is the transfer of heat from pan to food, or from direct heat source to food. The temperatures are typically much higher than for steaming (steam is at 212F, 100C), so you get entirely different chemical effects from the process (e.g., caramelization). Therefore, I doubt the fish would "fry" in its own oil in the case of steaming.

If you're concerned about maintaining the salinity of your brined fish, you could consider steaming it en Papillote, or even in foil. This would result in the fish cooking in its own moisture, essentially eliminating the leaching of salt (although some salt would move to the surface, carried by the water).