How does "purification" happen?
Based on what happens in the canon novel Ahsoka, crystals are purified by meditating with them; this is essentially the only thing Ahsoka does between the Inquisitor's (red) lightsaber exploding and her (white) lightsabers igniting:
Before she could even consider shouting a warning, the red lightsaber burst into a mess of noise and light. Bright spots pricked at her eyes, and then all was quiet. The creature wasn't going to bother her anymore.
She pushed aside her sense of urgency, even though she was in a hurry. Meditation came easily, as if she were sitting in safety in the Jedi Temple itself, instead of a dusty street with her enemies closing in. Her mind's eye sorted the preassembled components and those she had just retrieved into order, locking each into place with the others. When Ahsoka opened her hands, she was
not surprised to find that two lightsabers, rough and unfinished, were waiting.
They would need more work, but they were hers.
When she turned them on, they shone the brightest white.
Ahsoka Chapter 28
Presumably, within the Force this is Ahsoka creating a bond with her crystal; Pablo Hidalgo has said on Twitter that this is how crystals change colour for Jedi, which is one of the few things we know about the process:
@primagrine what determines a light sider's lightsaber colour? do the crystals [have] anythin[g] to do w[ith] it [?]
@pablohidalgo The crystal takes on a color when a Jedi achieves a bond with "his/her" kyber. No one knows what the colors signify.
On an internal level, "purification" would seem to be resolving the Dark Side/Light Side conflict within the crystal; Ahsoka suggests that crystals are turned red by tipping the scales in favour of the Dark (emphasis mine):
Dark crystals were made, too, but not in that holy place. They were plundered from their rightful bearers and corrupted by the hands that stole them. Even rock could be changed by the power of the Force, bleeding alterations until their color was the deepest red. The balance was finely staged between the two, light and dark, and it took very little to upset it.
Whether or not a crystal can be "purified" by anyone, or just by the person that crystal chooses as its proper bearer, is unknown.
Does "purification" always result in a white (or colourless) crystal?
We don't know; Ahsoka is the only example we have of this process in action, and her lightsaber colour explicitly symbolizes her detachment from both the Jedi and the Sith. Even Pablo Hidalgo doesn't know, having been asked this on Twitter:
@ReyaShankar Do red crystals turn only white when restored or can they change to other colors as well?
@pablohidalgo Unknown. In fact, Ahsoka was very surprised when it happened.
If forced to guess, I would assume that a "purified" crystal would turn to whatever colour was appropriate for the person who bonded with it; so, for example, Luke could have purified the crystal in Vader's lightsaber and turned it green. But this is, of course, only a guess.