No official cheat codes exist for BioShock on Xbox 360. The game was designed without a cheat console or button-code system.
PC version has BioShock HD / Remastered console commands (`F10` or `~` via ini edit) and mods (Widescreen Fix, FOV, Texture, Cheat Tables). Not transferable to 360.
Play on Easy + Vita-Chambers + Smart Plasmid/Weapon use = "God Mode" feel without cheats.