Boundary conditions are the top of the atmosphere and the Earth's surface/oceans. Partly, atmospheric climate model simulations are directly coupled to ocean model simulations. This makes the models even larger and harder to setup/run. "Forcing data" means thinks like CO2 emissions, emissions of particulate matter, solar radiation ... .
Having said that: it is quite unrealistic to setup und run a "real" climate model just for fun.
Simulation results of the couple model intercomparison project (CMIP) form the basis for the IPCC reports. The CMIP Phase 6 (CMIP6) simulations form the basis of the most recent IPCC reports. There is an overview paper of CMIP6 which might indicate how large the overhead for running "real" climate models in a comparable way is (Eyring et al., 2016).
One of the models that were used for the CMIP5 and CMIP6 simuations was the MPI-ESM model. An overview of this model is given in Giorgette et al. (2013). Detailed descriptions are available in this special issue.
There are several more models used in CMIP5 and CMIP6. The MPI-ESM is just an example.
There are some simple climate models that are made for students and which are based on a few governing equations. Seems models show the general features of climate models but can be run on a normal end-user computer. One of these models is the "Monash simple climate model". An instance of that model which can be run via a web-GUI is available here. There is a documentation available in which you will find links to a publication describing the model and to a repository to download the source code.
There is another climate model made for training which is called Planet Simulator. Download details and documentation are offered on that web page.
The last term is the energy moved from one band to another, in a very simple (read "crude") "diffusion" model. C is a constant, Ti is and temperature, and T is the average temperature of the whole globe (I'll tell you how to calculate that in a minute). Again, values for C vary. Recently I used C = 3.74 W m^-2 K^-1.
You can find the mean global albedo or temperature or what-have-you by multiplying the zonal values by the zonal area fraction and adding up all the figures. You find the fraction of a hemisphere's area by subtracting the sine of the more equatorial boundary from the sine of the more polar boundary. For instance, the area of a hemisphere between 30 and 40 degrees is sin(40) - sin(30) = 0.1428.
P. Chýlek, J.A. Coakley, Jr. J. Atmos. Sci. 32, 675-679 (1975).
S.G. Warren, S.H. Schneider. J. Atmos. Sci. 36, 1377-1391 (1979).