White balancing in the camera to an image on a monitor requires more than one exposure; all the methods have similar setup requirements:
Turn off power-saving and screen blanking software. Let the monitor warm up for half an hour. You don’t want the monitor to be changing during the process. Half an hour is probably overkill for an LED backlight, but it’s not too long for a florescent one.
Darken the room. You want your images to be repeatable, and the easiest way to avoid extraneous changes in the monitor is by eliminating any other source of illumination.
Put the camera on a tripod. If you take it off the tripod to read out the images, make sure you can put it back in the same place. A quick-release mount helps a lot here. Mark the place where the tripod’s feet go with tape in case you jostle or kick it during the calibration.
Line the camera lens up perpendicular to the screen. It doesn’t have to be perfect, especially for Newton’s method. For gridded search, lens-axis/screen orthogonality will reduce errors and therefore the number of iterations.
Use a fairly long lens. LCD brightness and color varies with viewing angle, and you want all the ray angles approximately perpendicular to the plane of the screen. A 135mm lens on a full frame camera, a 90mm lens on an APS-C camera, or a 70 mm lens on a four-thirds or micro four-thirds camera is long enough. If you can’t use a long lens, Newton’s Method is probably your best bet.
Defocus the lens. If you don’t defocus the lens, you can get moiré patterns that somewhat affect the color as the pitch of the camera’s pixels interacts with the pitch of the monitor’s pixels. You don’t have to defocus very far. If you use Newton’s Method (see below) you can defocus are far as you want. If you use the gridded search approach, don’t defocus so far that you can’t find the grid.
Use a longish exposure. You don’t want the screen’s refresh rate to influence the images. I suggest a tenth of a second or longer, although I admit that that’s conservative.
Stop down the lens a couple of stops or more. This minimizes corner falloff. This is quite important for the gridded search method, and not so important for Newton’s Method.
Use the base ISO setting. You want to minimize the amount of noise in your images. Setting the ISO control to the lowest setting that’s not a “pull” will minimize the noise. Don’t worry about your exposures getting too long; even if they’re a second or two, it’ll be just fine.
Use a calibrated monitor. This is not a firm requirement for some methods. Newton’s Method will probably converge faster with a calibrated monitor, and any starting points obtained from other people will depend on monitors having similar characteristics.