This achievement is triggered by having precisely 200 dog tags. The easy way I've found to do this is to use the cheat for "infinite dog tags" which gives you 999. You then spend them on continues until your total is below 200, after which you can collect enough to reach 200.
Note that this solution is of limited usefulness given the availability of other cheats. Most players will likely choose to collect 200 tags while playing through to unlock other achievements. However, as Bastian Reader
informed me, it's possible that the game won't spawn enough tags on your run to reach 200. So, if you're only after the dog tag achievements, you might prefer this method over another complete playthrough.
Access the cheat menu by highlighting "options," then press start while holding LT + Y + left.
Minimally, you want to enable infinite dog tags. I also enabled infinite grenades and ammo. Just be sure that you don't enable infinite health or continues because you need to be able to die and to buy continues.
The idea is to die intentionally so you can use continues. It's probably safer and easier to burn through continues at the boss of each level, but remember that level 3 doesn't have a boss. The final room on level 3 is recognizable by having a sealed door at the top of the screen.
You'll begin level 1 with 3 continues. If you choose, you can use those before completing the level. At the first upgrade screen, you can spend tags on all upgrades and max out your continues. Note that it might be worth skipping health upgrades to make it easier to die later.
From here on, you play through the levels, dying and using up your continues, then buying more at each upgrade screen. Remember, try to avoid collecting dog tags. When you finally spend enough tags to get your counter below 200, begin collecting until you hit 200. Using this method, I got the achievement on level 4.