Does anyone know a budget travel agency to arrange our 5 days stay in Beijing?

First wish you a happy time of your 5 days Beijing Tour(https://www.travelchinaguide.com/package/beijing.htm) . I suggest you one itinerary, hope it be helpful.
First day, arrive in Beijing and check-in your hotel, the walk to explore the city.
Second day, go to visit the Tiananmen Square, the Forbidden City and the Temple of Heaven.
Third day, visit the Badaling Great Wall, the Sacred Way and the Ming Tomb in the afternoon. On the way back to downtown, drive by the Bird's Nest (Olympic National Stadium).
Fourth day, visit the Summer Place, Beijing Zoo and Beijing Hutong.
Fifth day, do some shopping and leave to the airport.
Travelchinaguide is a good travel agency for your China travel(https://www.travelchinaguide.com/) . I used to join its tour which was very wonderful.