Kill Arkham Asylum is a perfectly serviceable prologue comic series that uses five issues to introduce the members of the Suicide Squad that feature in the Kill the Justice League game and shows how they got to where they are when the game begins.
Kill the Justice League had big shoes to fill, following on from the Batman: Arkham series, and it failed miserably. Fortunately, this series fares much better. The story is more interesting (and, unlike the game, occasionally funny or well-written), the characters are presented with more appealing personalities through issues dedicated to each of the four of them, and the art style balances the grit of the Arkham universe with the lighter tone of the Suicide Squad game's visuals. In particular, King Shark's thoughts and dialogue here blow the game's version out of the water.
Unfortunately, it still doesn't really feel like Batman: Arkham to me. The introductory issue tries hard to tie things back to the Arkham games, with Amanda Waller rebuilding Arkham Asylum, and there are constant references to events of the games throughout the series. But it still feels jarring to see Batman again and hard to believe it's the same character I spent so many games controlling.
Overall, this is a fine series that suffers from one fatal flaw: it's a prologue to a very, very bad videogame that you should avoid playing. And since that's the case, why read it?