Born on Monday

I find this character to be one of the most interesting villains, not just of Batman, but the entire DC comics shared universe. Unlike most Batman villains, this guy pops up as a threat to numerous Justice League heroes. 

Solomon Grundy was created in 1944 by Alfred Bester and Paul Reinman. First appearing in All American Comics #61, but not as a Batman opponent. Originally, Solomon Grundy is an adversary of Alan Scott, the Green Lantern of the Justice Society. But, both characters, Alan Scott and Solomon Grundy, are from Gotham City. Bester and Reinman base the character on the old British nursery rhyme of the same name, HP Lovecraft's Herbert West: Reanimator serial short, and folktales of revenants. The nursery rhyme is below, a verse that the character repeats often during his appearances in the comics. 

Solomon Grundy, born on a Monday,

Christened on a stark and stormy Tuesday,
Married on a grey and grisly Wednesday,
Took ill on a mild and mellow Thursday,
Grew worse on a bright and breezy Friday,
Died on a gay and glorious Saturday,
Buried on a baking, blistering Sunday.
That was the end of Solomon Grundy

His origin varies depending on the writer, however, the basic premise stays the same. Cyrus Gold, sometime in the 1890s, is murdered and dumped in Slaughter Swamp, right outside Gotham City. Fifty years later, he resurfaces as a reanimated zombie-like revanant known as Solomon Grundy. 

Not only is this guy a brutish, shambling undead monster, he's nearly immortal. Grundy is somehow connected to The Green, DC Comics's version of Mother Nature, which means, Grundy's powers are related to those of Swamp Thing, the protector of The Green. Every time Grundy "dies", he returns to The Green and is reborn. Each time, Grundy's personality and abilities are a bit different. 

Grundy makes appearances across the DC universe as a member of the Injustice Society, and the Secret Society of Supervillains. He shows up in Grant Morrison's Seven Soldiers of Victory mini series, Jeph Loeb's Long Halloween miniseries, and Hush storyline, Willingham and Sturges's Salvation Run miniseries, and the Blackest Night crossover, and has big parts in the Arkham City and Injustice videogames.