The overarching climate models also typically aren't individual monolithic models, but tend involve various multidisciplinary pieces spanning much of the physical sciences. Also, get ready to work in Fortran. I spent a summer at Lawrence Livermore National Lab, and met some people there whose work was mostly the uncertainty quantification of outside climate models. My understanding at the time was that the lab specifically does not produce their own model, but rather, acts as an impartial testing ground for the major efforts from other global institutions.
All of that aside, they do have plenty of problems, and people from the outside do come in and make improvements, it's just not necessarily easy to break into. If you wanted something more accessible to start thinking about, you could do worse than the edGCM from Columbia[1].
[1] http://edgcm.columbia.edu/software2/edgcm-climate-modeling/
As someone with a small amount of background in software engineering (though it's not my primary job) I can say that they do think about these things and do have very experienced software developers working on these projects, but help is always appreciated!
Speaking to the accuracy of the simulations I do think that's more in the domain scientist's court though, since most of these things use pretty specialized numerical models and physical approximations.
Edit: I'd also like to say that I pretty much agree with everything cwal said.
Also, feedback loops are something they have only very recently considered.
http://www.realclimate.org/index.php/data-sources/?wpmp_tp=1
The models are all numerical simulations. But even if you are skilled in numerical calculations you'd need a lot of energy to learn and understand the domain too.
Here's one university elementary-introduction-level course about that:
http://www.cs.toronto.edu/~sme/CSC2602/
and one book with the real formulas:
http://www.climate.unibe.ch/~stocker/papers/stocker14icm.pdf