You can check your progress on this in the awards section:
On your map, click
to open driver ID.
Press
until you reach "awards".
Go to the tab named "PvE" in the "MISC" section and open it.
Go to the "Mission A" tab and select it.
Third one in the bottom row shows "Man On A Mission" progress.
There are 65 story missions, all of which need to be beaten with a platinum medal if you're going for the following trophy:
So you may want to work on that before trying to grind out this trophy.
Unfortunately; no, faction missions don't count for this (I tried) so you can't get this while grinding for reputation points.