You are using an entry-level, consumer grade DSLR that has a phase-detection auto focus module having limited performance. It's not nearly as sophisticated as the AF systems in Canon's more expensive cameras. The AF module is actually from an older model and expert reviews give that AF module low marks.
Though your T3i (US), Kiss X5 (Japan), 600D (everywhere else) has 9 AF points, only the center AF point is a cross-type focus point.
If a person's eyes are in shadow cast by their brows (dark eye sockets/raccoon eyes) there may not be enough light for AF to work well, but the center cross-type point will out-perform the other 8 regular AF points. You may want to use the camera's AF assist light or a reflector get sufficient light into the subjects eye sockets to help AF.
Subjects shot outside often have dark eye sockets.
In Live View, because the main mirror up the 600D can't use the phase-detection AF module and has to rely on the less accurate, and in the 600D slow, contrast-detection AF system.
You apparently have only a partial understanding of what DoF is, how DoF relates to image focus, and how to control DoF to your advantage.
You're not alone. DoF is the most complex technical aspect of doing photography and the way it works is not as immediately intuitive as the exposure triad is.
The more you understand about how your DSLR camera's various subsystems work, the better you are able to get the camera to do what you want done.
Autofocus - Wikipedia, the free encyclopedia
Understanding Camera Autofocus
Tutorials - Sharpness
Understanding Depth of Field in Photography