The titular 'Dogs of Death' are the evil-spirits-of-the-week, they're old enemies of Monkey with a plan to kill him. One of them has the power to create plagues, so they put one in the way of the pilgrimage and Tripitaka, who can't resist the urge to help (even though he has no medical skills at all), catches it.
The only cure for the plague is "monkey's brains," and most of the episode is spent with Monkey agonising over whether he should sacrifice himself for Tripitaka, and then, when he decides he will, trying to kill himself.
Of course for Monkey suicide isn't easy, because he's indestructible (a fact seldom remembered about him in the series, but an important plot point here). He tries various methods, each more tedious than the last - sadly there's no dramatic tension in these scenes because (1) we know Monkey isn't going to die, and (2) it's attempted suicide played for laughs.
Eventually the head dog-spirit (who has been disguised as a doctor at the plague hospital) provides them with a magic axe that he claims will kill Monkey. He can't help gloating and gives himself away, leading to a climactic fight which is admittedly rather good, but hardly worth the wait.