In order to meet a clone, you need to complete a run at least once.
After this, clones will randomly appear on the sector map. I've never seen one appear in Sector 1, it's always Sector 2 onwards. They are marked by an icon with a DNA helix in purple.
You just need to jump into the area with a clone and kill it for this to pop.
In earlier sectors, they spawn with no shields and limited equipment/drones. In later sectors, they will literally shred you to pieces in a matter of seconds if you're not careful.