Make the Blackberry Simple Syrup:
In a small saucepan, combine the blackberries, sugar, and water.
Bring to a simmer over medium heat, stirring occasionally, until the sugar is dissolved and the blackberries break down, about 5-7 minutes.
Remove from heat and strain the syrup through a fine mesh sieve, pressing on the solids to extract as much liquid as possible. Discard the solids and let the syrup cool.
Prepare the Glass:
If desired, rim a chilled martini glass with sugar. To do this, rub a lemon wedge around the rim of the glass, then dip it into a plate of sugar.
Mix the Martini:
In a cocktail shaker, combine the vodka, lemon juice, blackberry simple syrup, triple sec, and simple syrup.
Add a handful of ice and shake vigorously until well chilled.
Serve:
Strain the cocktail into the prepared martini glass.
Garnish:
Garnish with a fresh blackberry and a twist of lemon peel on the rim of the glass for a finishing touch.