With respect to targeting this achievement, there isn't a huge amount of difference between the classes.
The one thing that may significantly impact your quest for this achievement is the fact that if your buddy dies, if you can revive them before the end of the fight, you're still on track for the achievement. What that means is that you may want to build your character and select skills in such a way as to minimize your use of debuffs or Damage Over Time effects. (This has the side effect, incidentally, of making much of the game much, much, harder, for what it's worth, so I'm not sure I recommend going all in on the strategy - easier to reload, IMO.) This will allow you to make combat a bit more predictable, at the expense of it lasting longer. Incidentally, the class best suited to avoiding the application of DoT's - the Fighter - is also the class with access to the widest variety of abilities that can lower your opponents Attack, which is also useful for pursuing this achievement.
Another option is to focus more on preventing foes from acting. In this case, the Mage has access to a Stun from right near the beginning of the game, and later, gains access to the only innate Slow effect - which can be very useful.
Finally, the Jew and Thief both have access to abilities that can apply 'Pissed Off' to targets; this functions as a taunt of sorts, forcing enemies to attack your character instead of your Buddy. (Do note, however, that the Jew gains access to this at a much lower level than the thief).
Really though, all of these differences are pretty minor. What matters more is your consumable use, and willingness to reload if things do go a little pear shaped. Every class has the ability to do lots of damage, and once you get to a high enough level to kill a mob in one attack, (the Sucker Punch perk helps) the Swat Helmet and some smart use of your 'Shoot' skill on the map will mean that you can end many fights before an enemy even has a chance to try to hit your buddy.
Best Answer
When Critically injured, your health bar will flash. It applies to all or most character in the game.