Getsu Fuma
Intially, Getsu Fuma (Fuma Getsu using the Western naming convention) had little relation to the Castlevania series. Although his game was produced by Konami in between the original Castlevania and Castlevania II, Getsu Fuma Den was set in a different continuity. It also never made it to the U.S. -- His game was heavily rooted in Japanese mythology, so it's easy to assume that Konami expected his story just wouldn't translate properly for Western audiences. Thus, outside of Japan, little was known about this hero at all.
This, of course, continued on for some time as his next appearance, as part of Konami Wai Wai World, also never made it to the U.S. Unless you were a hardcore gamer, one with a keen interest in Japanese releases, you probably never knew about Getsu Fuma at all.
That changed, though, with the release of Castlevania: Harmony of Despair, the big, mish-mash, multi-hero extravaganza Konami released for online play. When Getsu Fuma was included as a bonus DLC character, players finally got to meet the demon-fighting swordsman.
Character History:
Getsu Fuma Den
Once, many years ago, in the early days of the Demon Age, there were three Getsu brothers, skilled swordsmen dedicated to fighting demons. Together, these three wiedled powerful pulse blades, weapons capable of defeating demons with ease.
When the demon lord Ryukotsuki rose to power, he knew that the three brothers would have to be defeated. He put all his might into the battle with the brothers, felling two. However, the third and youngest, Fuma, escaped his death and fled to battle.
Regrouping, Getsu Fuma rededicated himself to defeating Ryukotsuki. He ventured into the hellish demon realm, traveling across many islands there and defeating the many demons he found. This allowed Getsu Fuma to power-up his blade, making it more powerful.
Once he had bested all the forces that hell could offer, Getsu Fuma went to the final land of Ryukotsuki. There, the two did great battle. Unlike before, though, Ryukotsuki was unable to easily fell the Getsu swordsman, instead falling to the might of the young man's blade.
Victory achieved, Getsu Fuma rested knowing that he has restored honor to his clan in avenging the deaths of his brothers.
Non-continuity History:
Konami Wai Wai World
The first Konami Wai Wai World follows the adventures of Konami Man and Konami Lady as they venture to various Konami worlds to rescue Konami heroes (every time you say "Konami" an angel gets its wings). Getsu Fuma is trapped in Getsu Fuma Den world (i.e., Hell). There the two heroes have to fight a great demon who has Getsu Fuma prisoner.
Once Dracula is defeated, though, Getsu Fuma becomes a playable character, and he's able to aid the Konami heroes in exploring the other Konami worlds and saving the universe of Konami.
Castlevania: Harmony of Despair
Getsu Fuma is one of the DLC characters in Harmony of Despair, in his first appearance directly tied to Castlevania continuity. Of course, this whole game is a flight-of-fancy, non-continuity adventure, so nothing that happened in the game actually had any effect on the greater continuity
Still, it's not hard to see that IGA has a fascination for the character, especially when he essentially based a new character, Zangetsu on this character in the Bloodstained series.
Playing as Getsu Fuma:
Getsu Fuma is a strong, but limited character. In his first appearance he's able to attack using his swords as well as via demonic powers (his version of subweapons, activated by pressing down-B). His range is limited with his main attack, though (the swords being a closer-range weapon than the full-powered Vampire Killer). His skills, though, are strong and deal a fair bit of damage (so long as Getsu Fuma has the magic to use them).
Along with side-scrolling sections that us Castlevania fans would be used to, Getsu Fuma's game also featured 3D-style maze exploration stages. These were lengthy, and controlled quite different from what we're used to. While impressive, these sections of his game did not make reappearances in his later cameo games.
Appearing in Harmony of Despair as a downloadable bonus character, Getsu Fuma's inclusion in the game was a bit of a fun lark, one of two 8-bit characters (the other being a retro Simon Belmont, for a nice nod towards Wai Wai World) throw into the otherwise very modern looking game. Here Getsu Fuma retains his swords and skills. Like retro-Simon, he's not necesarily as powerful as some of the other characters in the game, but he is fun to take for a run once in a while.