How does climb-rate vary with density/pressure altitude?
For propeller aircraft, the climb rate is a function of 


available power
demanded power
weight
air density
wing lift


Five variables, and wing lift is itself a function of Mach number, Reynolds number, wing AoA, wing area. Available power is a function of air density, throttle setting, propeller incidence - demanded power a function of air speed, air density, angle of attack, Mach &amp; Reynolds numbers. So in total a very large matrix of independent variables - in order to find equations via an analysis, we'll have to make some assumptions and simplifications. For instance that the aircraft thrust vector stays reasonably horizontal so that $T \cdot sin(\Gamma)$ is close to zero and can be disregarded. Also, that lift = weight during the climb.

For the steady climb, the weight equation then becomes $$W = C_L \cdot \frac{1}{2} \rho V^2 \cdot S \Rightarrow V = \sqrt{\frac{W}{S}\cdot\frac{2}{\rho} \cdot \frac{1}{C_L}} \tag{1}$$

For the drag in horizontal flight:

$$ D_h = C_D \cdot \frac{1}{2}\rho V^2 \cdot S = \frac{C_D}{C_L} \cdot W \tag{2}$$

and the required power in horizontal flight $(P_r)_h$ becomes:

$$ (P_r)_h = D_h \cdot V = W \cdot \sqrt{\frac{W}{S}\cdot\frac{2}{\rho} \cdot \frac{{C_D}^2}{{C_L}^3}} \tag{3}$$

The power required to maintain climb speed $C$ is $W \cdot C$ and available power $P_a = (P_r)_h + W \cdot C$, hence:

$$ C = \frac{P_a - (P_r)_h}{W} = \frac{P_C}{W} \tag{4}$$
Combine (3) and (4):

$$ C = \frac{P_a - (P_r)_h}{W} = \frac{P_a}{W} - \sqrt{\frac{W}{S}\cdot\frac{2}{\rho} \cdot \frac{{C_D}^2}{{C_L}^3}} = \frac{\eta \cdot P_{br}}{W} - \sqrt{\frac{W}{S}\cdot\frac{2}{\rho} \cdot \frac{{C_D}^2}{{C_L}^3}} \tag{5}$$



Above picture shows a graph of $P_{br}$ of the P&amp;W Wasp: function of manifold pressure and altitude. This engine had a turbo-charger for improved altitude performance, the engines of GA aircraft may not have these. Plots of variable pitch propellers show a propeller efficiency $\eta$ of about 0.8. 

How this ties into the graph shown in OP:


The equations feature air density $\rho$. This is a function of static pressure and of temperature: an equation to convert to static pressure and vice versa can be found here.
Available power for a normally aspirated piston engine decreases as a function of altitude, approximately according to $\frac{({P_{br}})_h}{({P_{br}})_o} = (1 + c) \frac{\rho_h}{\rho_o}$. Tests on some American piston engines showed that for many of them a value of $C$ = 0.132 would be appropriate, refer to figure below which also shows the altitude-power function of a piston engine with supercharger.




All references and pictures from a university lecture book, paper copy only.