The Fire Count was a villain who appears in "His Hero". He wears a crown which suggests that he may have been the previous ruler of the Fire Kingdom. He is made out of fire and has nothing else inside of him but his facial expressions. He can be considered a "representation" of Ice King in the sense that Billy is Ooo's old equivalent of Finn. Considering that he is a count and thus royalty, it is likely that he is related to Flame King and Flame Princess in some way. In his appearance, Billy fought him and rescued the Cotton Candy Princess. He was bisected by Billy, so it is doubtful he will appear again any time soon (aside from flashbacks).
His face slightly resembles Ice King's in that he has a goblin-like nose and sharp teeth. Otherwise he looks like a classic fireball, except inverted so that the rounded part of the flame is towards the top and the tip of the flame is at the bottom.
The Fire Count was originally named the Fire King. Some say the change was made very late in the production of "His Hero."
Unlike Billy though, Finn does not desire to slay the Ice King. Probably he thinks Ice King is only crazy and misunderstood, and should only be taught a lesson.