The pistol is available to all classes and is the easiest one to get if you start working on it right away. No matter your class, put in enough points into the Pistol tree so that you can get the Marksman skill, as it is immensely helpful for this achievement. When you get the skill, use it as often as you can, as it makes using the pistol very effective. Even if you don't play as a Soldier or a class that doesn't have any training for that particular weapon, it is still very possible to get the achievement. Focus on getting as many accuracy improving attachments for the guns you aren't trained in, and stabilizers for Sniper Rifles. It's not as good as playing as a class trained for the weapon, but it's better than nothing. Also, kills made by teammates do not count towards this achievement.
Note: Sentinels don't have pistol training. They get pistol bonuses from their class skill.
All weapon kills carry over between playthroughs and careers. Even if you don’t get all 150 kills in one playthrough, any kills you have made will carry over into every other one you may have.