Solomon Grundy has a few different origin stories. I believe the name (but not the origin story) is inspired by the 1842 English nursery rhyme of the same name:
This poem is quoted in the game, but the origins of the character differ. Initially, in the late 19th century, a wealthy merchant named Cyrus Gold is murdered and his body is disposed of in Slaughter Swamp. Fifty years later he comes back as a revenant type supervillain and occasionally anti-hero. He is initially a villain for Alan Scott's Green Lantern but crosses over and becomes a problem for other heroes. He's basically unkillable with both super strength and stamina. He seems to be partly made of the same life force as Swamp thing and is nearly invulnerable to fire, cold, magic and other energy attacks.
In Arkham City his origin story is a bit different: first appearing in Batman: Arkham City, this version was a merchant who was murdered and had his body dumped in a swamp-like Lazarus Pit, where a strange thunderstorm combined with the pit's mystical properties to resurrect him as an effectively immortal zombie-like monster. He temporarily became part of a travelling circus before he was found by Ra's al Ghul, who realized the swamp's true nature and took him to Wonder City. Seeking to understand the pit's properties, Ra's repeatedly murdered Grundy until Wonder City was eventually shut down and Gotham City built over it, with the Iceberg Lounge being built over Grundy's prison. After the Penguin finds him, he uses Grundy to kill traitorous underlings and attempts to kill Batman, who eventually defeats Grundy.