You'll need a pot (not a pan, something deep enough to submerge the parts), a metal mesh strainer, soap, toothbrush, disposable wooden chopsticks (if you know how to use them ) or a disposable fork will work too, and finally the dye (of course).
Prep your figure by thoroughly scrubbing with laundry soap or dishsoap (I use the toothbrush to get in the cracks). Rinse off and make sure to shake and rinse to get all the soap out.
Pour the dye (either powder form or liquid) into the pot along with some water, stir and bring it to a high temp (but not boiling!). Once the water is hot enough, turn the heat down to as low as you can. Stir again.
Place all the parts you want to dye in the strainer, and dip the strainer into the pot. The strainer prevents the parts from resting on the bottom of the pot and possibily warping (or worse, melting the parts). I bent the handle on my strainer, so that it hooks over the side of the pot and rests there not touching the bottom, so I don't end up holding the thing for the duration of the process.
Now here's the variable. Some parts take a lot longer than others to dye, so you'll have to check for yourself. Generally speaking, softer plastic absorbs the dye a lot faster and hard plastic takes more time. Leave the parts submerge in the dye, occasionally stirring the pot with the chopsticks or fork, and turn the parts. Do this every 10~15 mins. This can take 45 mins to 1.5 hours usually.
When you feel like the parts have achieved the shade that you want, lift up the strainer with all the parts, run them under the faucet and after the residual dye washes off, see if they are indeed the color you want. The water removes the excessive dye that hasn't been absorbed, so even though in the pot it looks great, the dye may not be absorbed as much as it appears.
If more color is needed, put the parts with the strainer back in the pot and repeat.
Now the last step is very important.
After the parts have achieved the color you are happy with, fill a new clean pot with just water and bring it to a high temp (again, not boiling). Redip your parts into the clean water (using the strainer) and let the excessive dye bleed into the clean water. Do this at least once. I do it several times. You can also run the dyed parts under hot water (but this will take a lot of water). Basically, you want to rinse the parts until the water runs clear.
The reason being is that the plastic has absorbed a lot of dye, and if you're planning on painting details, or sculpting over the dyed parts, that dye color WILL resurface. So if you dye your parts red, and you paint the details white, yellow, etc, those parts will become a shade of red in a couple of weeks. If the details are painted blue, it may turn purple. I learned this the hard way after making a custom Snake Eyes - I painted all the straps and details a shade of grey, and had him displayed on my shelf. A few weeks later, the grey all disappeared, and he was standing there, in solid black....
Hope this helps.