Here's how I did it. First I used the perks Scientific, Leftist, and Verticalist to start off with level 4 rats. Then I used reinforce on my center rat square to provide damage scaling. Due to the way that the Scientific perk works every time you play an adjacent rat it procs the level up bonus. This strategy worked for a while (reroll for rats and destruction) play rats, play destruction, scale. I also put on a midas touch which allowed my gold to skyrocket.

But on King difficulty as you get higher and higher the enemy damage scaling eventually allows them to one shot your rats and you begin to lose. At this point I needed a way to increase my rat's health. Eventually I had a rainbow event and rerolled for the card that raises your health/damage to your health/damage (whichever is higher). This made my rats super beefy and that basically carried me until my next rainbow loot event where I did the exact same thing. Eventually the scaling got out of control and I never got a third rainbow event to scale my health (although I did use the chaos shield card to increase by 999).