For the puree: Add the blackberries to a medium saucepan with the sugar and lime juice. Cook over low heat, covered, 20 to 25 minutes. Strain using a fine mesh strainer, pressing the berries to extract as much juice/puree as possible. Place in the fridge to cool completely.
For the lime sugar: Mix together the sugar and lime zest. Use a piece of lime to moisten the rim of each margarita glass, and then dip in the lime sugar.
For the margaritas: In a blender, add 1 cup of the tequila, 1/4 cup of the triple sec, 1/4 cup of the sugar, and the juice of 1 lime. Fill the blender with ice and blend until smooth. Then add as much blackberry puree as desired, 1/3 to 1/2 cup. Pour into 4 rimmed glasses and serve immediately. Repeat with the rest of the ingredients for the other 4 glasses.