I would suggest installing a curtain rod specifically for hanging the blackout curtains instead of trying to attach them to anything on the vertical blind mechanism. Attaching the curtains would likely prevent any operation of the vertical blinds at a minimum, and possibly damage the mechanism.
The new rod could be above the vertical blind, or, if the rod is long enough, the brackets could be at the same level as the blind mechanism and the top of the curtain essentially flush with the top of the opening. This would probably depend on how wide and tall your curtains are.
Doing it this way would allow you to use your existing curtains and to continue to use the vertical blinds when you do want light and to continue to use the sliding door that I presume is behind the blinds. You would have to spend for the curtain rod, but if money is tight, you can get a very inexpensive plain rod. If you'd like to spend a bit more, you can get something fancier and more decorative.
If you're renting and worried about damaging the wall, usually a few screw holes aren't enough to cause loss of a damage deposit. Or you can buy a small container of drywall spackle to easily patch up the holes before you leave.