What is the influence of Classical antiquity to Italian renaissance and neoclassical art? (especially portraits)
This is a big question so the answer has to be rather general. However, here are some things to consider. First of all, the Renaissance was obsessed with antiquity. What many artists and other thinkers sought was a return to and reconsideration of the works of the past, often enhancing or elaborating on those works.
Two artist you may want to consider for further study are Botticelli and Donatello. Bottecelli's paintings featured a "revival of mythology within the focus on antiquity." Donatello's sculptures of saints were based on those of ancient war heroes.
In architecture, Giovanni Medici poured money into projects revering the past. One such effort was renovating the Church and the Sacristy. Medici believed “the ancient world provided symbols of timeless stability.”