How can I calculate maximum rate of climb?
To calculate your possible climb speed $v_z$, you will need


Your engine's thrust $T$
Your airplane's drag $D$
Your airplane's mass $m$


Calculate how much power is needed to overcome drag, and any excess can be used for climbing: $$v_z = v\cdot sin\gamma = v\cdot\frac{T-D}{m\cdot g}$$

Note that this equation makes use of several simplifications, but works well for propeller and slow turbofan aircraft with moderate flight path angles $\gamma$.

To do this with more precision, you need to account for the fact that the aircraft should accelerate during the climb to stay at the same polar point. Now you further need:


The gradient of air temperature over altitude (lapse rate $\Gamma$)
The local speed of sound $a$, and
The gas constant $R$ of air.


You need to add a correction factor $C$ which has several components: $$C = 1 + \frac{1}{2}\cdot\kappa\cdot R_w\cdot\Gamma_w\cdot Ma^2 + \frac{(1+0.2\cdot Ma^2)^{\frac{\kappa}{\kappa-1}}-1}{(1+0.2\cdot Ma^2)^{\frac{1}{\kappa-1}}}$$

where $\kappa$ is the ratio of the specific heats of air and is 1.405, the index w denotes the wet adiabatic gas constant and lapse rate of air, and $Ma$ is your flight Mach number. $\Gamma$ can vary between -0.004°/m and -0.0097°/m, but if you use the average of -0.0065°/m, this equation can be simplified to: $$C = 1 - 0.13335\cdot Ma^2 + \frac{(1+0.2\cdot Ma^2)^{3.5}-1}{(1+0.2\cdot Ma^2)^{2.5}}$$

which is probably the form you will find in most books which care to cover this topic. The second summand takes care of the reduction of atmospheric temperature with altitude and disappears in the stratosphere, and the third summand covers the additional energy needed for acceleration in terms of flight Mach number.



Acceleration factor over the flight Mach number in the troposphere for the Standard Atmosphere

Now your climb speed becomes $$v_z = \frac{v}{C}\cdot sin\gamma = \frac{v}{C}\cdot\frac{T-D}{m\cdot g}$$

As you can see from the graph above, the correction factor is only important at higher speeds, but will cut the climb speed in half at Mach 2. Some jet aircraft need to climb at a high constant Mach number, and then the aircraft needs to decelerate while climbing. Now the correction factor becomes smaller than unity and the climb speed gets a boost because kinetic energy is converted into potential energy while climbing.

Optimum speeds

To pick the flight speed where the climb speed or the fight path angle reaches a maximum, you now need to describe how thrust will change with flight speed. To simplify things, we can say that thrust changes over speed in proportion to the expression $v^{n_v}$ where $n_v$ is a constant which depends on engine type. Piston aircraft have constant power output, and thrust is inverse with speed over the speed range of acceptable propeller efficiencies, hence $n_v$ becomes -1 for piston aircraft. Turboprops make some use of ram pressure, so they profit a little from flying faster, but not much. Their $n_v$ is -0.8 to -0.6. Turbofans are better in utilizing ram pressure, and their $n_v$ is -0.5 to -0.2. The higher the bypass ratio, the more negative their $n_v$ becomes. Jets (think J-79 or even the old Jumo-004) have approximately constant thrust over speed, at least in subsonic flow. Their $n_v$ is around 0. Positive values of $n_v$ can be found with ramjets - they develop more thrust the faster they move through the air.

The flight speed for maximum rate of climb ($v_y$) is reached at a lift coefficient $c_L$ of $$c_L = -\frac{n_v+1}{2}\cdot\frac{T\cdot\pi\cdot AR\cdot\epsilon}{m\cdot g}\cdot \sqrt{\frac{(n_v+1)^2}{4}\cdot\left(\frac{T\cdot\pi\cdot AR\cdot\epsilon}{m\cdot g}\right)^2 + 3\cdot c_{D0}\cdot\pi\cdot AR\cdot\epsilon}$$

whereas the steepest climb is possible with a $c_L$ of $$c_L = -\frac{n_v}{4}\cdot\frac{T\cdot\pi\cdot AR\cdot\epsilon}{m\cdot g}\cdot \sqrt{\frac{n_v^2}{16}\cdot\left(\frac{T\cdot\pi\cdot AR\cdot\epsilon}{m\cdot g}\right)^2 + c_{D0}\cdot\pi\cdot AR\cdot\epsilon}$$

Nomenclature:
    $c_L \:\:\:$ lift coefficient
    $T \:\:\:\:$  thrust 
    $m \:\:\:\:$  aircraft mass 
    $g \:\:\:\:\:$  gravity 
    $\pi \:\:\:\:\:$  3.14159$\dots$
    $AR \:\:$  aspect ratio of the wing
    $\epsilon \:\:\:\:\:$  the wing's Oswald factor
    $c_{D0} \:$  zero-lift drag coefficient