If its only approximate that you want, then the rate of climb is estimated by - power produced minus power required for level flight at the same speed.
E.g. If you have 200HP available at 100 knots and 100HP is required to maintain level flight at that speed, then the surplus 100HP is available for the climb. 1HP will raise 550lb at 1 ft/s, so 100hp will raise e.g. 2200lb at 25 ft/s or 1500 ft/min.
So basically at the same speed, rate of climb is inversely proportional to weight (not quite true as drag will increase with weight).
The calculation for different speeds at the same weight isn't quite as simple, but you could approximate the drag as being proportional to the square of the speed (i.e. constant Cd) and estimate the change in level flight power requirement for the two speeds and correct your RoC in proportion to the change in surplus power.
If you have flight manual data for the power required (and available) at different speeds and weights, then you could refine the calculations accordingly.