Preheat the oven to 350F. Add cupcake liners to muffin pan.
In the bowl of your mixer, add butter and sugar and mix on high until well blended and fluffy.
Turn down mixer to medium speed and add eggs, one at a time, beating after each addition.
Next, turn down mixer to low speed and add in flour, lime zest, baking powder, baking soda and salt.
Lastly, add in lime juice, milk and vanilla extract and mix until well blended.
Using an ice cream scooper, fill each cupcake liner 3/4 full and bake for 20-22 minutes or until a toothpick inserted in the center of a cupcake comes out clean.
Place cupcake pan on cooling rack for 10 minutes.
For the Blackberry Buttercream:
Clean your stand mixer bowl and whisk attachment. Add the confectioners' sugar and butter and mix on low speed, increasing the speed to medium-high once the sugar is fully incorporated. Continue to mix until light and fluffy.
Add the blackberry puree, vanilla extract, and salt and beat until the mixture is puree is fully incorporated.
Once the cupcakes have cooled completely, pipe them with the buttercream.