If you wash the fat- as you are cutting it into chunks to run through a grinder, you can get rid of much of the "salt", and if you soak the fat in cold water, much of the blood will leech out. The cleaner the fat is before it goes into the grinder, the less unwanted blood, sinew, hair and salt you have to separate from the oil.
Using a double boiler system- a pot in a pan of water- with water in the pot that holds the fat-- will keep you from burning the fat, starting fires, and provides an efficient way to transmit heat to the fat, to get it to "melt".
If you have enough fat- as this poster has bear fat-- you can easily spoon or pour off the oils from the top of the water into a separage container, to cool. I think the idea of grinding the fat to get the most oil out of it is important, but in the long run, you have to filter, and distill the product several times to get clear oil. I have used coffee filters, as well as cheesecloth folded in several layers and placed in a strainer to separate out junk from the oil. But the only way I know to get the salts out is by using water with the oil so that the salt leeches into the water. Several Changes of the water during the rendering of any large quantity of oil is called for.
When to change? If the water appears " bloody", Change it! When it cools, you can do a taste test to see how salty the water is. If you can taste a strong taste of salt, change it for new, fresh soft water. Cooled, the oil rendered from the fat will either congeal, or will at least float on top of the remaining water. Either way, the oil can be separated from the waste water, and then reheated to process it again.
I too have speeded up the process by putting a contained of water and oil into my freezer, to congeal the oil and force crud down into the water, which will also freeze. Then remove from the freezer, spoon, or scrape the grease off the ice, and repeat the heating/rendering process in a new container with water. Discard the frozen " water" portion, and clean the container for continued use.
Once you remove the solids from the first batch of rendered oil, I have found no need to continue to attempt to screen out "crud" with coffee filters and/or cheesecloth. Its gone, or its too small to be screen by anything I have on hand. I have to rely on precipitation and absorption processes to clean the oil further. Cleaning the water several times seems to be the only short way to refine the grease to oil.
