To ‘act the part’ is to behave as others expect you to behave so they’ll attribute to you what you want them to attribute to you.
Examples:
- Walk confidently, wear business casual, and carry a clipboard
- Lie, in such a way that you won’t be discovered, about your past to make you appear more desirable/important
- Pretend to have more authority/confidence/competance/power than you do
Related: Fake it ’till you make it.