Death King boosts your revive bonus
Death King is a boss that improves the attack damage you gain per revive, according to gamestratwiki, with the effect reportedly scaling at higher boss levels. No source checked publishes the exact bonus curve, so the improvement size at any given level is unconfirmed.
Cursed King increases damage taken — on purpose
Cursed King raises the damage you take, which sounds like a downside, but gamestratwiki notes this actually speeds up your death-to-revive cycle, since dying faster means looping through the core progression loop faster. It's a boss that works with the game's core mechanic rather than against it.
Where you encounter King Bosses
Both bosses are encountered while progressing through the Death Road and Death Tower, per gamestratwiki. Neither boss has a confirmed fixed location or level requirement in the sources checked for this page.