I don't really recommend making these at home. The amount of work that is involved is not for the casual breakfast cook. Nonetheless, this is what you do:
I happen to have a cheese grater that can shred a potato really finely. Once shredded, I plopped my potatoes into a bowl and poured boiling water from the kettle over them. This "blanches" them and removes some of their starch. After blanching them, drain them and squeeze them dry. You want them dry so that the patty doesn't splatter when you drop it into very hot oil. Grease burns on the forearms are not for the home cook.
Salt the potatoes and form the patties. Easy. Because they are so finely shredded, your patties should hold their shapes without the need for a binder. Right. That's done. Now it's time to cook them. Twice.
Using the same technique I use for french fries, I first deep fried the patties in medium hot oil. If you haven't blanched them enough, the remaining starch will quickly turn your patty dark brown, like molasses color. OK, pull the potatoes out of the medium hot oil when they're getting a little color and set them out on a paper bag. Then crank up the heat under the oil as high as you possibly can and finish your patties. When they look like perfect hash brown patties, pull them out and set them again on a paper bag to sop up some of the oil.
Apply ketchup. EAt.