Place the blackberries, mint leaves, lemon juice and simple syrup in a cocktail shaker and muddle to break up the berries and bruise the mint leaves.
Add a few pieces of ice (3 or 4, roughly) and the bourbon then shake to mix and chill.
Strain the mixture into a whiskey glass with a few pieces of ice in it and top up with soda water.
Video
Notes
If you don't have a cocktail shaker, you can always improvise and eg use a sealable jar or measuring jug with a lid, then the end of a wooden spoon or rolling pin to muddle (see video for my improvised set up!).