"Begone, Knave!" allows cards that deal Battle Damage to return to your hand.
This card can combo well in a Deck of Gemini monsters, which tend to have three things: lots of Level 4 or lower monsters, monsters with one-shot effects, and lots of cards to Special Summon from the Graveyard.
Use this on "Dark Valkyria" after using her effect as a possible means of reviving her for re-use later.
Use this on "Gemini Soldier" to activate its effect on your turn without taking damage and protect your Life Points.
Use this on a monster equipped with "Supervise" to revive a Normal or Gemini monster from your Grave after the Battle Phase.
This card works very well in a Koa'ki Meiru Deck. Most Koa'ki Meiru are Level 4 monsters with high ATK, and end up destroyed during the End Phase anyway. Also, by using it with "Core Overclock", you can deal enormous damage with a single monster.
This card works very well in a Dark Scorpion deck, allowing them access to their effects.
As its own effect destroys it anyway, "Infernal Dragon" is an excellent choice to use this card on, and has the potential to cause an enormous amount of Direct Damage. "Keeper of the Shrine" can be used to provide some defense afterwards.
Or you can activate "Deck Devastation Virus" immediately after attacking with it, allowing it to cause more harm to your opponent before the Battle Phase even ends.
A good idea is to use this card on a monster while you control "Wattpheasant". Attack directly with the targeted monster and then with "Wattpheasant", using "Wattpheasant" to banish the other monster. The monster will return during the End Phase, avoiding the effect of "Infected Mail" that would send that monster to the Graveyard.
This card can work very well in a Gladiator Beast deck, as it allows you to attack even when your opponent's monsters are stronger than yours, and your monsters will be returned to the Deck before they are sent to the graveyard by this card.
In a Banish deck, you can use this card with "Gren Maju Da Eiza" to inflict very large amounts of damage to your opponent.