You don't do "Wanted: Alive or Alive
" in a mission, it while you are "free roaming" with Trevor. It is triggered by completing a certain mission, however. As long as Trevor is alive, you can do this. Same goes for any achievement that requires a certain character.
Most of the achievements are done outside of missions anyway. There are some, like getting 70 Gold medals, need to be done in the missions because it's based on your performance in each mission. For those, yes, they can be replayed. Plus, only the secret achievements are story related. Pretty much every other one is done in-between missions while free roaming.
I'm sure you took a look at the guide before asking this, but in case you didn't...:
Achievement Guide & Roadmap