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There are numerous ways to get a 3-star rating, because there are numerous ratings in the first place.

The most obvious way is simple: Get a long kill streak. A 3 kill streak earns one star, a 6 kill streak earns two, and a 9 kill streak earns three stars, at which point you get an achievement. Though load-out choice certainly comes down more to personal preference and play-style than anything, a loadout I'd recommend, which is available early on: Diablo SMG w/ Scope, Flashbangs, Flak Jacket (when going for a long kill streak, very useful) and Personal UAV (not a very team-player thing to do, but helps you get kills), Tactical Reload and "Straight from the Hip". As for play-style suggestions: stick to the edge of the map when you can, and slowly work in a circle. Sticking to the edge means that there's just that much less of an area the enemy can blindside you from, and working in a circle ensures you won't get caught in the same place by an enemy out for revenge. Plus, if you keep moving, you're less likely to be sniped. But these are just suggestions, feel free to do whatever works best for you personally.

Vehicle kill streaks are tracked separately, but follow the same scheme: 3 kills in one vehicle is one star, 6 is two stars, and so on. However, note that I said "in one vehicle", and not "in vehicles in one life". If your vehicle blows up, or you hop out, your streak will reset. So watch out. If you're looking for tips here, odds are you aren't good with vehicles, so the best suggestion for this that I can give is: Whenever the spawn option let's you pick "Teammate vehicle", take it. You may get lucky and get the passenger seat of an Apache. If your pilot's any good, you should be able to sponge enough kills off him/her to get a 3-star (or higher).

There are two types of drone streaks. Assault drone streaks (which function the same as other kill streaks, increasing in increments of three), and Parrot marking streaks, which start at 5 marked targets, but require 10 to get to the next level. (This is easier than it sounds though, because marking targets only requires you be in range, you don't have to kill anyone. Plus, the reticule for marking is pretty big, so you can mark several enemies at once.) When going for this achievement in the Parrot, it is ESSENTIAL to have the "my buddy" perk, as it greatly increases your battery life. I also would suggest "thick skin" and "drone be gone".

There are other streak types, but I haven't played the game enough to feel confident enough to post about them. If you're aware of any other streak types, the amounts needed to get a new star, have a good load-out for getting kills or... Well, anything, please post it in the comments so I can add it.

Also, as always, if I made any factual errors, please point it out so I can change it. Thanks!

EDIT: Tip for getting 3-stars in the Parrot: You can re-mark a target that has a blinking diamond and it will count towards your next star. So if you can find a particularly ignorant sniper and hover just a few feet behind him, you can repeatedly tap "RT" for marks, while staying in reasonable cover. I've gotten to 4-stars several times online just off of one guy doing this. (Bonus: if you have Crater-to-Order on, you get the bonus of very possibly killing the sniper once somebody realizes they should destroy your drone.)

2nd EDIT: If you don't have a "battle code", you can still get this achievement: Have a friend who does have a battle code pull you into a match with him/her, and you'll be able to get the achievement.