How to perform a "first principle" performance calculation?
The equations of motion are the easy part. In essence, you look at all forces affecting the aircraft (lift, thrust, drag, weight) and balance them with proper control settings (elevator, throttle) and accelerations (if thrust > drag, the forward acceleration is (thrust - drag)/mass).

This you repeat over and over, one timestep at a time. The next timestep sees the aircraft at a new speed, which you get by multiplying the forward acceleration with time, and altitude if the climb speed is nonzero. The new, changed mass is the old mass minus the fuel consumed during the last timestep. And so on. This involves coordinate translations as some forces are defined in the aerodynamic and others in the airplane coordinate system. NASA Langley has published an open source software which does exactly that (LaRCsim).

For very high precision you can even model the inertias and calculate which aileron deflection is needed to arrive at a desired roll rate in the next timestep, but even without that you will get very precise data if the forces are correct.

The hard part is to arrive at the correct forces. We have had several questions here asking for the aerodynamic data of modern airliners, and every time the answer was: They are kept secret. You need to do your own analysis, and it is the same with engine data. Older methods relied on tabulated data, but in order to calculate conditions outside of the validity of the table they need to calculate the forces analytically. To get an idea what parameters need to be considered, look at this answer about the Boeing SCAP module.

However, even some crude assumptions can get you very close to the real result.