The straightforward method is to use the software unless you're a glutton for punishment and have lots of free time on your hands. :-)

There's much more to monitor behavior than just the full-on red, green and blue response in isolation. A monitor's picture controls don't give you enough control to correct for what is often nonlinear behavior across all shades, which means there's no way you can produce accurate color with the method you describe. The best you could hope for is colors close to those you measured being right and others being wrong. Because of this, useful profiles are the result of displaying and measuring a lot of colors and contain a large lookup table.

The usual process is to put the monitor in a known state and produce a profile of its behavior in that state. The profile is then loaded into a program that can do the compensation on its own, into the OS if that's supported or directly onto the video card where it's done in hardware. I've never seen a monitor that can be given an externally-produced calibration; if such a thing exists it's the exception rather than the rule since the other methods are much more practical. Since the profile has to be put into a file to be used, there's not a lot of reason to avoid having software involved in the process.

The profiling process can be understood in detail by studying the source code and documentation for ArgyllCMS and DisplayCal. Both are the result of a lot of work figuring out what needs to be done and codifying it. Understanding why monitors perform as they do would be off topic here, but I'm sure plenty has been written about it. If you're still inclined to develop a profile manually, follow the same process that the software would use and you'll end up with enough data to produce one.

Most lab instruments worth their salt can be controlled by a computer, and the manufacturers usually provide documentation of how that is accomplished. ArgyllCMS supports many devices, including a couple of lab instruments. If you have the background to add support for yours or can team up with someone who does (maybe visit the Computer Science department and ask), doing so would benefit you and anyone else who might want to use the same device.