Preheat oven to 350 degrees F.
Line 2 standard cupcake pans with 24 liners.
In a large saucepan over medium heat, simmer coconut rum and butter.
Slowly whisk cocoa powder into saucepan until mixture is creamy.
Remove from heat and allow to cool.
Meanwhile, whisk sugar, flour, salt and baking soda in a large bowl.
In a separate bowl, beat eggs and greek yogurt with an electric mixer. Slowly add coconut rum and cocoa mixture. Combine on low speed.
Slowly add flour and sugar mixture, combining on low speed until completely incorporated.
Fill baking cups three-fourths full. Bake for about 22 minutes. Cool.
To make the frosting cream butter until smooth. Gradually add powdered sugar, alternating with rum and vanilla, until desired consistency is reached. Feel free to add more or less sugar or liquid based on how thick you want your frosting.