The industry standard for first-person view simulation in most shooters is to have character models and animations distinct from those used for third-person view. There are several reasons for this:
The player has a much smaller field of view upon the world than a real person in the character's situation would, and he lacks other forms of input such as feeling sensations.
Since the game must obey the usability principle of giving feedback for all of the player's actions, a lot of actions and objects must be displayed as if in front of the player character's eyes. E.g. when hip-firing for real, you don't see the gun at all or barely but you can feel it, whereas in an FPS you see it in plain view. When reloading a gun for real, you must look down at your pocket to retrieve the mag (or not if you can just run your hand down your vest or something) and then look down at the gun where you place the mag, whereas in an FPS, this is done in front of the player character's eyes.
The third-person model is not detailed enough to be displayed from up close. If it were used for first-person view, the low resolution of the texture and the poor geometry would hurt the player's eyes... Using an LOD is not the solution because you have to redo the models anyway due to the previous bullet point.
edit: I forgot to add this interesting link.
Finally, again due to the field of view being narrower in games than in real life, the game must somehow display objects that are below the character's head as higher than they really are for the player to get a better understanding of their relative position. Indeed, when you look straight in reality, there's a whole range that is outside of the standard 75-90 degrees FOV in games, where you can waive your arms with your eyes able to see them, whereas in a game you can't. In a game, what happens in front of you must be displayed.
This is done by lowering the camera a bit with respect to its realistic position about the eyes.
There is no universal rule for how much lower than the eyes the camera has to be, but I guess you can try somewhere between the shoulders and the nose (that's a rough estimate), see what's best for you and that should do it.
If you can play Counter-Strike or COD in multiplayer with a friend, try to see what happens when you make your characters face each other and toy with the cameras. I might try that myself and post a screenshot.
edit 2: in games where there's a button to level your eyesight ("center view" button), if you approach a standing NPC, apparently in most cases you'll be looking at the base of their neck. See these planetside 2 screenshots.
edit 3: also, shoulder-high is where the weapon actually is on the character model, so if the camera is there too then bullets always travel along a straight line from your "eye" to what your crosshair is on, which is pretty convenient and, in some of the most hardcore FPSes, downright essential to gameplay.