How can I be a police officer if I am a green card holder?

Eligibility to be hired as a police officer is not governed by U.S. immigration laws, but instead is governed by the laws and regulations of the police force's locality.

It just depends on where you want to work. Some cities or states will not hire non-citizens, but many will hire green card holders. You would need to check with the specific police department in your area to see if they hire green card holders. If they do, then find out the specific requirements to be hired.