If you experience elbow pain or sore biceps after doing strict pull-ups or strict muscle ups you may be using too much of your bicep and not enough of your lat.
You wouldn't choose to do a bicep curl with a dumbbell of your bodyweight, yet so many athletes rely solely on their biceps in the pull-up and/or muscle up. Your lat is a much bigger and stronger muscle than your bicep.
Your lat also allows you to pull the rings with a greater range of motion. This is highly important when we talk about the pull of the muscle up. It's easy to notice an athlete who doesn't have enough lat engagement in their pull because of his or her end range of motion. The athlete who pulls from bicep engagement will only have an end range of motion where the elbow meets the shoulder line and the hands cannot reach the sternum. The athlete who pulls with lat engagement will have an end range of motion where the elbow goes past the midline, behind his or her body, and the hands are able to reach the sternum.
Here is an example. Watch when her elbows surpass her body and she is able to pull the rings down to her sternum:
The main reason athletes use their biceps rather than their lats is due to poor lat activation in the pull. Lat activation happens even before you bend your arms; activation starts when you retract your scapula. When you are in a dead hang on the bar or the rings before doing a pull-up, the first instinct is to bend the arms thus stressing the biceps. To activate the lats before you bend your arms, you must begin your pull by getting a good false grip then retracting your scapula. This is critical when activating the lats in the pull on the bar or the rings.
Retracting the scapula can be harder than you would think. Practice retracting from a dead hang on the pull-up bar, rings or push-up position. Remember not to retract upwards and put stress on the traps or neck. Once you feel comfortable just retracting, add a push downward. Push down on the pull-up bar or rings without bending your arms. Again, the lats engage before you even bend your arms so if you find yourself unable to push down on the equipment without bending your arms, it may simply be an issue of coordination. You probably won't get very far when pushing downwards on the equipment, but you feel your lats doing the work.
Finally, remember that any compensation you teach yourself will eventually have to be corrected. Focus on coordination and activation of the correct muscles.