For tourists from most non-Latin American countries, all you need to enter Peru from its official entry points with Ecuador, Colombia, Brazil, Bolivia, and Chile is current and valid passport.  However, if you have been residing in any of these countries for more than 90 days, it is expected that you will present an official "permission to leave" slip, usually obtained at your host country's immigration office.  Due to Peru’s limited public service budget, none of the terrestrial entry ports are open 24/7, so it is best to plan your arrival at these borders accordingly.
A word of caution: the moment you get off your bus or step out your taxi, any number of locals will aggressively "offer" you help, and explain all sorts of "fees" and "taxes" you will need to pay, and how they will serve as your guide. They will also willing to change money for you.  Use common sense and deal only with recognizable and verifiable border guards and immigration officials, and change your money only at banks or other legitimate businesses.