Dead Goblin made a brief appearance in the episode "The Eyes." The presence of this goblin creates a "goblin problem" for Peppermint Butler, who states: "I swear I found him like this, I don't know what happened."
The outward appearance of Dead Goblin strongly indicates that he is indeed dead. His skin is a pale greyish blue color, which is very different from the varying shades of green that most goblins display. He seems to be in a state of emaciation, as his rib cage is highly visible. It could also be possible that he was dehydrated. His ears and nose are broken and wrinkled.
He is also on the tall end of the scale, even when the varying heights of most goblins are taken into account. His head is roughly the size of Peppermint Butler's entire body, and his body is longer than Finn's body and head put together. (Finn, Jake, and Peppermint Butler seem to be only a foot or so farther from the viewer than Dead Goblin, so his perceived size is not likely due to his proximity to the viewer.)
Circumstances of Death
Dead Goblin is shown lying in a shallow puddle of water, evidently spilled by a nearby overturned wooden bucket.
The residence that Dead Goblin is seen in front of is in an advanced state of dilapidation. (The windows are broken, the molding of the door frame is peeled off to some degree, and a large chunk seems to be missing from one of its walls, among other indications of decay.)
Both the architectural style of the house and the pink trees in the background could indicate that his death and discovery took place inside the Candy Kingdom.
The goblin's rib cage is also visible, indicating that it could have died from lack of food or dehydration.
Peppermint Butler himself may have been responsible for his death, due to his dark, mysterious past.
Circumstances of Discovery
During a flashback montage, Jake reminds Finn of their time spent supposedly disposing of Dead Goblin's corpse at the request of Peppermint Butler.
The exact exchange that the viewer is shown is as follows:
Peppermint Butler (As he leads Finn and Jake to Dead Goblin): "I swear I found him like this, I don't know what happened."
Finn: "We'll take care of it, Peppermint Butler."
The conversation is similar to the various "corpse disposal" situations that occur in mafia movies or in several episodes of The Sopranos. In those situations, one of the characters does something violent and irrational, such as spontaneously killing a fellow mob member or a prostitute. After coming to his senses, the character then desperately needs help getting rid of the evidence to avoid prosecution. To avoid suspicion by eavesdroppers, the character tells his associates that he needs help "solving a problem."
Knowing a bit of Peppermint Butler's back story and violent tendencies makes this suggestion more credible. Jake and Finn likely help Peppermint Butler out of naive generosity, instead of attempting to cover up a crime.