'Soft-Resetting' just means restarting the game without quitting out first.
- Save in front of the legendary Pokemon
- Start the battle
- If it's not shiny, press and hold L+R and press either Select or Start to perform a soft-reset.
- The screen will go black and the game will start loading the title screen.
- Some legenaries are still shiny-locked (At this stage believed to be only Groudon, Kyogre, and Rayquaza), therefore you cannot get them Shiny.
- The chance of encountering a shiny is really low, and no different between regular wild Pokemon and legendaries.
- It's a 16/65536 (1/4096) chance of finding one, or 0.0244%.
In response to your edit:
What you are effectively doing by soft resetting is dice-rolling until you get a certain number, except in this case the dice has 4096 sides.
To automate the process you would need everything from a program that can tell whether a Pokemon is shiny, to somehow automatically pushing the buttons (using robotics or directly soldering your 3DS). Needless to say, automation is not really an option unless you're willing to do some heavy modification to your 3DS. See BlueRaja's answer for options to do so
If you are adamant about saving time, you could change the Pokemon to be shiny after its caught using a Powersave/Action Replay device. Do note that this is directly cheating and generally frowned upon by the wider community (so is soft-resetting according to some purists but it's less frowned upon than Powersave).