The ritual purification is done by the deacon, instituted acolyte or priest (in that order, if present). It can be during or immediately after Mass.
The credence table is preferred (GIRM n. 279). But by a priest it may be done at the altar or credence table. By a deacon or instituted acolyte only the credence table is mentioned.
After the ritual purification, after Mass, there can be cleaning and polishing of the chalice. This would normally be done by the sacristan in the sacristy.
From the 2002 General Instruction of the Roman Missal (GIRM) approved for the USA, which can be accessed from romanrite.com/girm.html :
“183. When the distribution of Communion is completed, the deacon returns to the altar with the priest and collects the fragments, if any remain, and then carries the chalice and other sacred vessels to the credence table, where he purifies them and arranges them in the usual way while the priest returns to the chair. It is also permissible to leave the vessels that need to be purified, suitably covered, at the credence table on a corporal and to purify them immediately after Mass following the dismissal of the people.”
“192. Likewise, when the distribution of Communion is completed, a duly instituted acolyte helps the priest or deacon to purify and arrange the sacred vessels. When no deacon is present, a duly instituted acolyte carries the sacred vessels to the credence table and there purifies, wipes, and arranges them in the usual way.”
From n. 163 (Mass without a deacon):
“Upon returning to the altar, the priest collects any fragments that may remain. Then, standing at the altar or at the credence table, he purifies the paten or ciborium over the chalice then purifies the chalice, saying quietly, Quod ore sumpsimus (Lord, may I receive), and dries the chalice with a purificator. If the vessels are purified at the altar, they are carried to the credence table by a minister. Nevertheless, it is also permitted, especially if there are several vessels to be purified, to leave them suitably covered on a corporal, either at the altar or at the credence table, and to purify them immediately after Mass following the dismissal of the people.”
“279. The sacred vessels are purified by the priest, the deacon, or an instituted acolyte after Communion or after Mass, insofar as possible at the credence table. …”.