The most relevant sections of the Catechism of the Catholic Church seem to be 662 and 664. The first focuses on Christ's priestly ministry:
Jesus Christ, the one priest of the new and eternal Covenant, "entered, not into a sanctuary made by human hands. . . but into heaven itself, now to appear in the presence of God on our behalf." There Christ permanently exercises his priesthood, for he "always lives to make intercession" for "those who draw near to God through him". As "high priest of the good things to come" he is the centre and the principal actor of the liturgy that honours the Father in heaven.
Through the prayer of his people, Jesus is in communion with his body, and acts as their priest, hearing their prayers and interceding on their behalf. (cf. CCC 2565)
Section 664 deals with Christ's rule over his kingdom:
Being seated at the Father's right hand signifies the inauguration of the Messiah's kingdom, the fulfilment of the prophet Daniel's vision concerning the Son of man: "To him was given dominion and glory and kingdom, that all peoples, nations, and languages should serve him; his dominion is an everlasting dominion, which shall not pass away, and his kingdom one that shall not be destroyed." After this event the apostles became witnesses of the "kingdom [that] will have no end".
Thus, according to the catechism, the two primary activities of Christ during his session are a) intercession as our high priest and b) ruling over his kingdom.