Do what you would with playing Iceberg Lettuce, since they are practically the same. However, Big Chill gives you a card, unlike Iceberg Lettuce.
You can either play this on turn 1 for a card advantage, as well as stalling an early threat, or consider it as an extra Iceberg Lettuce and play it much later. You can also activate the abilities of plants that benefit from freezing zombies, such as Snowdrop and Winter Squash.
Big Chill can also be used with Bananasaurus Rex as it is a cheap way of boosting it. However, only Green Shadow can use this combo normally.
However, watch out for Quickdraw Con Man, as this superpower will always force you to take at 1 Bullseye damage for each Quickdraw Con Man.
Against
Because Iceberg Lettuce is very similar to Big Chill, any strategy against Iceberg Lettuce applies to Big Chill as well. Using Quickdraw Con Man can also force your opponent to take 1 damage if they obtain this superpower.