There is more to focus than that unfortunately. Focusing an auto focus camera is an art in itself and takes some understanding and practice.
An auto-focus camera focuses on contrast and detail, and if there is more contrast or light in your background, the camera tends to go there for a focusing point.
Focusing on eyes is fine for close up stuff, but if you are a wee bit back from the subject the camera sees much more than just the eyes, so picks whatever has the most contrast in the focal area. For example the image
below . . . the camera would see approximately the amount in the circle to choose a focal point. As you can see, the area with the most contrast isn't the eyes, even although that is what is in the centre of the circle and
is what YOU want to focus on. It would probably go for the very high contrast of the stocking hung behind the subject. That would result in the eyes being out of focus.
Shooting is a three step process with auto-focus. Push the shutter release half way down and the camera focuses (and sets exposure if you are in an automatic exposure mode). The 2nd step is to recompose. The 3rd step is to push the shutter release the rest of the way to fire the shutter.
For every picture I take, I decide on a point on which to focus, in an area that doesn't have a background which could compete for attention. (For example I might focus on a white shirt and tie that are about the same distance from the camera as the eyes that I want in perfect focus) Then I
recompose the picture in my viewfinder while still holding the shutter release half way down . . then shoot.
Every shot . . . always . . . is focus . . . recompose . . . shoot. Sometimes I hold that shutter release half way down for up to a minute waiting for the right expression before shooting.