Are you not getting enough kills in the Crucible? Are you impatient, and is that why you're not getting enough kills? Well, STOP! noob. The shotguns is definetly right for you.
1st Tip:
Short range is key. Why did I skip what weapons you should use, because all the shotguns can kill you. Common like Preacher mk. 22 to exotics like 4th Horseman. Unless you're holding a Felwinter's Lie (Which you probably aren't), you gonna have to hip fire. Maps like the Anomaly (DUH) are great for shotguns.
2nd Tip:
Another fun fact about the Anomaly, for all the Christmas Noobs (Including Myself): It was only a Skirmish Map, until the newer updates came in.
Vertical or Sliding. By sliding or jumping, you have a better chance to kill the player before they kill you, even if they have shotguns also. Running in like crazy will get you some kills, but more deaths as a result.
3rd Tip:
Now the special ammo drops. You will run out of shotgun ammo, and if you do and don't know the spawns, you're basically screwed. Most good players know where the drops are. That's why sometimes, it's not okay just go on Rambo and try to kill as much players with little ammo.
4th Tip
Perks. Since every shotgun can just kill anybody, the perks makes it different. Rangefinder helps on shotguns that have a very small range. Cascade can also work, but is not the best. Full auto would also work, and is better than most perks you get. So GET THE RIGHT PERKS.
Those tips can help you win carefully and not like Rambo. If you want to feel dirty, then use the shotgun.