To calculate the drag force on any object including a rocket heat shield, you need to know the flowfield and thus skin friction distribution over the heat shield. For a realistic answer, usually numerical solutions to the Navier-Stokes equations are obtained to compute the boundary layer flow and skin friction distribution over the entire heat shield. With this knowledge, one can easily integrate the skin friction distribution to obtain the associated drag coefficient and force for the heat shield. Granted this is just the viscous component of drag, generally with rockets you will also have a fairly large component of what is called wave drag. This is the drag caused by flying supersonic and a consequence of essentially pulling along a shock wave with the vehicle.
At the conceptual stage of design, a pencil and paper (usually integrated into a simple code) is used to estimate the drag coefficient on certain components of a rocket. A very common technique is to look at the body as a two-dimensional object, and compute the inviscid flowfield around this object. Depending on the Mach number you are flying, there are many different panel method techniques for doing this. With knowledge of the inviscid flowfield, you can estimate the boundary layer flow properties using the Thwaites-Walz method. This is an empirical recast of the von Kármán integral momentum equation into a more useful empirical formulation. This method only works for incompressible flows. However, engineers typically extend this method to include compressible flows by using the Eckert (1956) reference temperature method. Basically, by evaluating all the thermophysical properties of the gas at a temperature defined as $T^*$, and using this within the Thwaites-Walz method, you will obtain surprising good agreement with compressible experimental data. Oddly enough, the reference temperature method was considered nearly top secret during the WWII era and paved the way for the success of the V2 rocket nozzle. Anyways, this will provide you enough to estimate the compressible skin-friction distribution over a two-dimensional body (representation of the heat shield) to calculate (estimate) the viscous drag. Obviously there is wave drag as well, but this can easily be obtained by classical aerodynamic methods.
At the more preliminary stage of design, researchers and engineers will usually jump ship from the reduced order analysis like Thwaites-Walz, and begin using government developed legacy codes like HABP (Hypersonic Arbitrary Body Program) and CBAero (Configuration Based Aerodynamics). These methods are all based on numerous different panel techniques but for 3-dimensional bodies. This will provide higher-fidelity analysis and estimates without making the final jump to the costly and timely analysis of full CFD. Only in the detailed design stage will researchers ever look into solving the Navier-Stokes equations and to my knowledge, this almost always is done using RANS (Reynolds-Averaged Navier-Stokes) with an algebraic representation of the Reynolds stresses.
Hope this helps.