Part 1: Gereral Overview

Resolution dependence and independence

At thier most basic visualized level, 3D models are made up of vectors consisting of vertices, faces and edges all of which are resolution independent. This allows you to zoom in as close as you want to your character, render it at any resolution, and yet see no pixilation occur.
However as soon as you apply a bitmap texture to your character it becomes resolution dependant.

This means that if you applied a bitmap texture of 256 x 256 pixels to your characters head, but rendered your character at 1024 x 1024 pixels (with your character's head encompassing the full resolution gate), your 3D application is going to have to stretch or interpolate your characters bitmap texture to four times its original size in order to accomodate for the missing pixels. This will ultimately result in the render output appearing as pixilated or blurry.

Output Resolution

Each of these output formats can have different target resolutions. For example, if your final product was for Full HD, we know that the final output resolution would not exceed 1920 x 1080 pixels. Therefore when it comes to rendering a frame of your character, your 3D application's renderer could be set to match the final output resolution. In this case that would be 1920 pixels in width by 1080 pixels in height.

It is therefore safe to deduce that, if you had a wide shot of your character, your entire character would not take up anymore than a maximum of 1920 x 1080 pixels within a full frame. Bearing this in mind it stands to reason that your texture map, for your entire character in this particular shot, need not be much higher than the given resolution.

Whereas if you had a close-up shot of your character's head for the same output then the texture map for your character's head alone, would not exceed much more than 1920 x 1080 pixels.
Recalling that in the previous example your character utilized the same sized bitmap for its entire body. This also means that one should never feel limited to using one texture map on a character or character component. Texture maps can be created for the character on a per shot basis (more often) for non-interactive projects, this also means that the model should be versitile enough to reflect as wide a range of differing resolutions of texture maps as possible.

On the other hand when your final output is for a realtime rendering engine (eg. 3D games) utalizing multiple texture maps for a single character will often result in increased memory usage for the device and is therefore generally not the preffered approach to texture mapping in this particular situation.

Of course if there was a rule book to determine the ideal texture resolution based on output there would be no need to have this discussion, so it's worth bearing in mind that these are merely suggestions on appraoching a solution.
A table follows of suggested bitmap resolutions for certain outputs on a per shot basis .

FORMAT	OUTPUT RES	ASPECT RATIO	SUGGESTED MIN RES
PAL TV	720 x 576	4:3	1024 x 1024
HD	1920 x 1080	16:9	2048 x 2048
UHD	3840 x 2160	16:9	4096 x 4096