fulfill all 613 commandments

 However, it is well known that R. Isaac Luria, of blessed memory, stated10 that [the soul of] every11Jew needs to be reincarnated many times, until he will fulfill all 613 commandments of the Torah in thought, speech and action, i.e., using all three soul-garments with which one performs the commandments.

להשלים לבושי נפשו ולתקנם שלא יהא לבושא דחסרא כו'

This is in order to complete the garments of his soul and to correct them, so that no garment will be incomplete.

As explained in Tanya, ch. 4, the divine soul has three garments - the thought, speech and action of the 613 commandments of the Torah. If the performance of one of these commandments is lacking, the soul's garments are incomplete. It is thus necessary for each soul to perform all of the commandments.

לבד מצות התלויות במלך שהוא מוציא כל ישראל כי הוא כללות כולם כו'

Excepted12 are the commandments incumbent upon a king,13 because he discharges the obligation of all of Israel, as he is a corporate collective of them all.14

All Jews are incorporated within the king. When he performs those commandments that depend upon him, it is therefore considered as if all the Jewish people had performed them. As to the other commandments, however, if in one incarnation an individual did not perform them all, he must be reincarnated so that he will be able to do so.

Why must one have the garments of all 613 commandments?

והטעם הוא כדי להלביש כל תרי"ג בחי' וכחות שבנפשו

The reason is, in order to garb all the 613 aspects and powers of one's soul,