How can the soul be purified?

The purification (tazkiyah) of the soul has two meanings: The first one is to justify the soul, to consider it faultless, to cover up its mistakes and if possible to show them as virtues.  In the expressions above, purification is used in this sense.

The verse Justify not yourselves (an-Najm, 32) is in this sense.
In Risale-I Nur Collection, the state of an unpurified soul is described as follows:
One who loves himself if his evil-commanding soul has not been purified-will love no one else. Even if he apparently loves someone, he does not do so sincerely, but only for the pleasure of it and the benefits he receives. He always tries to make himself liked. Also, he does not ascribe faults to himself; he defends and exonerates himself like a lawyer. He praises himself, exaggerating and even lying, showing himself to be free of fault, as though sanctifying himself. (Flashes)

There is also the purification of the soul that is encouraged by the verse: Truly he succeeds that purifies it. (ash-Shams, 9). Our scholars explain it as freeing the soul from evil deeds, that is, being freed from polytheism through belief and from sins through piety and then increasing it by good deeds.