Warm, moist air rises from sun-heated ground, cools as it expands, and the water vapour condenses at one exact height: the flat cloud base. How high the cloud grows depends on the temperature profile above it.
Orange: thermal air. Blue: surrounding air. Green dashed: thermal dew point. Cloud base is where thermal and dew point meet; the top is where the orange line crosses back below the blue one.
Sweeps T minus Td from 0 to 30 K at the current air temperature and compares three answers: a numerical parcel lift, Bolton's closed formula and the pilot's 125 m per kelvin rule.
The science
Dry adiabatic cooling. A rising unsaturated parcel expands without exchanging heat: T = T0 (p/p0)Rd/cp, about 9.8 K per km. The water-vapour mixing ratio r = 0.622 e/(p − e) is conserved, so the vapour pressure falls with p while the saturation pressure es(T) (Bolton: 6.112 exp(17.67 T/(T + 243.5)) hPa) falls faster.
Lifting condensation level. The height where e = es. This page finds it numerically on the hydrostatic pressure profile, then compares with Bolton's formula TLCL = 1/(1/(Td − 56) + ln(T/Td)/800) + 56 (kelvin) and with Espy's rule of thumb z ≈ 125 m × (T − Td).
Above the base. Condensation releases latent heat, so the parcel cools along the pseudo-adiabat dT/dz = −g (1 + Lv rs/(RdT)) / (cp + Lv² rs ε/(RdT²)), roughly 4 to 6 K/km in warm air. The thermal rises while it is warmer than its surroundings; buoyancy B = g (Tp − Te)/Te, the integral of B up to the top is the CAPE-like energy, and the maximum updraft is about √(2 CAPE).
Simplifications, stated honestly. No entrainment or mixing with the environment (real cloud tops are lower), no virtual-temperature correction, no ice phase or precipitation, ground pressure fixed at 1000 hPa, thermals start only with a temperature excess and the same mixing ratio, and the thermal's own cloud base is a little above the LCL of the surrounding air when the excess is non-zero. Moist-adiabat values around 4.7 K/km at 15 °C are textbook-typical, not measured here.