I approach this problem by thinking of what can lead to blurry images, then eliminating those factors.
Given your parameters, you will be sometimes shooting in low light with a long lens. That means opening up the aperture (reducing depth of field) or reducing shutter speed (increasing the chance of camera shake or subject movement), both of which can make it harder to get sharp portraits.
A high-quality stabilized lens, pro-level camera with many focus points and good high-ISO performance, good handholding technique (ideally a tripod but that won't work for active subjects) and knowledge of the DOF at any aperture/focal length/distance combination are all useful.
Practice your focusing and handholding technique. Since you want to use natural light, learn how to find the most and best illumination (open shade outdoors, window light with reflector fill indoors). Avoid low-light situations or use artificial light if it's dark.
Focus and recompose is fine IF you know that your initial focus point is in the same plane as the eyes. Remember that lens DOF is not flat, it's curved.
Finally, don't just shoot a couple of frames. I personally shoot a lot of images so we have a lot to choose from. It's easier to get a hero pick from fifty exposures than five.