Hello Flying for Fun,
All the information you're looking for can be found in FAR23.66 "Takeoff climb" and the other nearby paragraphs. If it is an earlier aircraft, CAR 3 standards might apply, but they are quite similar. FAR23 should be easily found on various websites. It is certainly on the FAA website, but not easy to find.
Do not expect to just go out and perform such a test. It requires a lot of precision to get an accurate result, and would be UN-NECESSARILY UNSAFE to test close to the ground. If the plane otherwise flys in trim, seems to handle properly, and has proper control travel available, the only other consideration for poor performance would be engine(s) not developing adequate power. Climb testing of any kind is probably the least effective way to diagnose engine power problems. Most of that kind of investigation is done on the ground by maintenance staff.
Cheers, Pilot DAR