“Persona Christi” applies to priests. When acting in his priestly functions, the priest is acting “in the person of Christ.” This means that in a mysterious way, through the sacrament of holy orders, the priest allows Jesus to make use of his his very being–his person–to accomplish sacramental actions. So that when the priest says, in the confesional, “I forgive you your sins,” he uses the word “I” but it is Christ acting through him. When he says, at Mass, “This is my body,” “This is my blood,” he uses the word “my” but it is Christ acting through him to effect the consecration.
“Alter Christus” on the other hand, means “another Christ.” And we are all called to be an “alter Christus” to one another.