The first thing you will need to do is learn how to turn and jump at the same time. there is two ways to do this.
The two ways to do this are switch the controls to stick and move or learn how to claw. Doing that will help you because you wont need to move your thumb of the thumb stick.
Next you will need to do is learn how to use the surrounding.
In Black Ops III there are thing called head glitches they only show you head making it harder to die.
The final tip for getting better at call of duty is using your movement.
In Black Ops III the game is set in the future where there is jet packs use them to your advantage by adding verticality to your movement.
Wall running is also in the game. On one map you can wall run from tree to tree on the forest.
How to jump and turn at the same time
How to use head glitches
How to use boosts and wall running