Should I run my condensate pump after pouring bleach water into it for cleaning?
The condensate pump should have a float switch in it so that it automatically comes on when there is water in it, and turns off when it's empty.

It is also possible to run the pump by pressing the test button (if it has one) or shorting the terminals on the float switch (if exposed) or in numerous other manners. Normally, you would not do this. However, if you do for whatever reason, then it is in these circumstances that you do not want to run the pump dry.

In your case, I would let the chlorinated water sit for up to 30 minutes, then pour in as much water as needed to completely flush the chlorine out of the well, pump and discharge line. Otherwise, if the pump has rubber seals the chlorine may damage them.