The term started being popularized by the game Unreal Tournament where the voice of the announcer shouts: GODLIKE! For all to listen, when a player kills 25 enemy players in a row, without dying. This feature was later added to other popular games, DotA-like games being of the most prominent.
Interesting Fact: In DotA you only have to kill 10 players in a row to get Godlike'd. If you kill more than 10 in a row, the DotA announcer says: HOLY SHIT! and adds a subtitle: 'Player is beyond Godlike. PLEASE, somebody kill him!'.
Mortal #1: Did we all just lose against this one guy?
Mortal #1: WTF just happened?
Mortal #2: Hello there,... are you god?
/* Unreal tournament voice: GODLIKE! */