We're talking about a dog that cannot die and wears people clothing, so you may already be asking how in the heck do you get an inugami of your own. Which also means that you didn't read the disclaimer to this entry, so here's your punishment.
There are a few ways to create an inugami. If you want to try them ALL, start by taking a few dogs and having them fight to the death. Then, you take the "winner" and bury it in the ground so that only its head sticks out. Now comes the starvation part. It's important to put food down juuuust outside the dog's reach so that it suffers mentally and physically while starving to death. Right before it expires, though, you cut off the dog's head in a way that makes it fly towards the food, which it still can't eat on account of the, you know, death. We're starting to suspect that old-timey Japan just didn't like dogs in general.
After the deed is done and God is done crying, the onmyodo sorcerer would burn the dog's body and keep the head, which anchored the angry dog spirit to Earth, and also stopped people from visiting his house. The inugami could then be made to kill the sorcerer's enemies and perform other tasks (other than the one thing all dogs live for: bringing joy to your life.) The only good news, which is a very relative term in a story like this, is that if you weren't super careful, the inugami spirit would turn against you and bring you untold pain and suffering. Still a far cry from what you'd really deserve (i.e. flesh-eating anal fleas) but it's better than nothing.