Training efficiently is something I always care about when I train myself as well as during personal training sessions. If you are not sure, I am here to help you.
What are the primary muscles for pullup / chinup?
Latissimus dorsi is the one you may think of first. Also your elbow flexors are the primary muscle for pullup / chinup. Depending on how you hold (overhand, underhand, semi pronated), targeting which elbow flexor muscle is different. And your posterior deltoid is the primary mover as well.
Engage the core muscles
When you perform bodyweight training, your body position and core engagement is significant. Pushup is also the one the core engagement is crucial. (You want to keep the lower back neutral all the time)
How should you perform pullup / chinup?
When you perform pullup / chinup, starting from a hanging position, pack your shoulders (depress and retract), then pull your body up. As your body goes up, your upper body won’t be straight vertical but slightly tilted.
What I sometimes see is that the lower body is hyperextended which is not the spine’s favorite position.
Is this not only good for your lower back health, but also you are missing the great chance to train your abs while doing pullup / chinup. In fact, pullup / chinup should be performed like the image below to optimize your training.