How do you make custom clothing change when the character is a different body weight? For example, I changed the size of the body using showracemenu but the size of the clothing stayed the same. I have two different sizes marked with _0 and _1 but I fail to see how to implement the different body sizes in the creation kit.

You have to have two separate and different (in terms of volume) meshes available for the game to morph them to the weight slider. If your _0 and _1 meshes are the same, you're not going to see any change in the weight.

Check your models and make sure that your _1 mesh is bigger than your _0 mesh. If not, you'll probably have to adapt it to the _1 body size.

In the CK you just add the _1 mesh.