Others have made some good points. There are many words for "acting like another person" with different connotations.
To "impersonate" or "pose as" means to pretend to be someone else with the intent to deceive, usually for some illegal or unethical purpose. Occasionally "impersonator" is used to describe someone who goes to great lengths to look and act like another person for entertainment purposes, like "Elvis impersonator" and "female impersonator". But I think those are rather special cases, we don't normally talk about an entertainer being an "impersonator".
To "play" someone is to pretend to be that person for a movie or stage play. Like you might say, "Bob Smith played Winston Churchill in a movie about World War 2."
To "imitate" is to copy someone else, but not to really pretend to be that person. "Imitate" is probably most often used to describe copying one particular behavior. Like you might say, "This famous athlete works very hard. I hope that children see that and imitate him." You don't necessarily want the children to wear the same clothes he does or talk like he does, etc, you just want them to copy this one behavior. It can also be used when the copying is just for amusement, like "George imitated Fred's walk" or "George imitated Fred's French accent".
To "mimic" or "ape" is to copy in a way that makes fun of the person. Like a bully is harassing someone and she says "please leave me alone" and so in an exaggerated whiny voice he repeats "please leave me alone". "Mimic" is also used as a technical term in biology to refer to a creature that resembles another creature, usually resulting in scaring off predators. Sometimes comedians who specialize in acting like famous people for entertainment are called "mimics".
To "parody" is to copy but not exactly, for humorous effect. You might say, "He took that serious song about undying love and made a PARODY of it about his love for hamburgers." "Parody" is also used metaphorically to mean that something is a copy that ruins the intent of the original. Like, "The biased judge and paid witnesses made the trial a PARODY of justice."
The most common phrase for what you are describing is probably "do an impression of", as in, "Bob is doing an impression of Marsha". In context, this is often shortened to simply "do", as in, "Hey, Bob is doing Marsha's morale building speech again", meaning he's doing an impression of Marsha giving this speech, probably making fun of her. Or a professional comedian might say, "Next, I will do Angela Merkel" to introduce his next impersonation.