Do I choose who to kill for Cull the Weak?

Yes, you do. According to the official rulings and FAQ for Cull the Weak, you are responsible for choosing which of your own units to kill when the spell resolves.

Here is how it works:

• No Targeting: Cull the Weak does not target any units when it is played. Because it does not target, no choices are made when you add the spell to the chain.
• Resolution Choice: The instruction "Each player kills one of their units" is performed during the resolution of the spell. At that time, each player—including you—independently chooses one unit they control to kill.
• "Do As Much As You Can": Per Rule 359.3.e.11, if you have no units at the time of resolution, you simply do not kill one, but your opponent is still required to kill one of theirs if they have any.

In short, you make the choice for your own units during the spell's resolution.