I'll make an attempt at this although it seems unclear whether you're asking (a) the theoretical ramifications of a certain chamber pressure or (b) the thermal and mechanical aspects of making it occur inside the chamber properly.
Before jumping into how one develops $p_c$, one should rigorously define it. This answer gives a good definition - that it is the static pressure within the chamber measured at or near the injector/chamber interface. In other words, it is the static pressure of the propellants immediately downstream of the injector, when they are first introduced to the engine.
In terms of how one arrives at a desired $p_c$, many of the answers are correct in that it is not a calculated parameter, but rather one which is arrived at by desiring a certain degree of performance. In general, increasing $p_c$ also increases $I_{sp}$ (albeit asymptotically) and decreases combustion chamber size. Increasing it also allows for larger nozzle ratios at a given exit pressure. These considered, there is a great impetus to run $p_c$ values as high as realistically achievable.
The determination of desired $p_c$ may be assisted by analytic tools such as NASA's Chemical Equilibrium with Applications or Ponomarenko's RPA, but there aren't (at least to my knowledge) straightforward means of calculating a $p_c$ operating point from performance levels. It is usually iteratively decided by setting a target, estimating performance, and adjusting. Certain values can be ruled out in specific cases by knowing other properties (i.e. RP-1 may go "sooty" below a certain pressure), but again, this is the land of "no general analytic rules" - many engine designs receive their first estimates from what has worked for similar engines in the past.
In terms of mechanically arriving at that pressure, once that value is set, one can define the factors used to develop the propellants to that pressure. In general, for pump-fed engines, one can simplistically model the propellant circuitry as a number of pressure drops*:
- The pressure drop of the injectors for fuel and oxidizer
- The pressure drop of the regenerative cooling jacket, if applicable
- The pressure drop of any ductwork and manifolds required to communicate the propellant between these areas
*This assumes a gas-generator cycle, as the pressure layout is most straightforward of all pump-fed engines. For staged-combustion and expander-cycle engines, the flow path is more complex.
In this case, the required outlet pressure of the propellant turbopumps equals the sum of the drops and the chamber pressure:
$$p_{op} = p_c + \Delta p_{inj} + \Delta p_{rg} + \Delta p_{\text{duct}}$$
Of course, this can quickly delve into a chicken-and-egg game whereby certain values which are not known require other values which may or may not be known. In that spirit, authors such as Sutton have developed rules of thumb for what these values should be - the general guidance for gas-generator cycles, for example, is that pump output pressure should be between 135% and 180% of $p_c$ (Sutton, 9e., Table 6-6).
As that allows definitions for $p_{op}$, one can now shape the other components (within limits, of course) to achieve the proper balance - the ductwork diameter can be driven to produce a certain $\Delta p_{\text{duct}}$, the injector orifice areas can be modified to give a desired $\Delta p_{inj}$, and the regenerative cooling channels can be sized similarly. It is intuitively reasonable that one wants these pressure drop taxes to be as low as possible, as this decreases both the required power of the pump and the mass of components. In practice, however, this may require a significant bit of iteration to make sure that reducing $p_{op}/p_c$ margins do not stress the mechanics of the overall system too greatly.
