How exactly are game theoretical evolutionary models described during implementation for computer simulations?
The field most closely associated with game theoretic models in biology is evolutionary game theory. If modeling is required, then the typical paradigm is agent-based modeling, and a good introductory book is:


  Yoav Shoham and Kevin Leyton-Brown
  [2009], "Multiagent systems:
  algorithmic, game-theoretic, and
  logical foundations", Cambridge
  University press.


As for actually building the model, and what to describe/how, I will take you through my usual procedure since this is a field I specialize in:


Define what type of strategies you think are relevant to the interactions you are modeling. Select what you expect the payoff of those strategies to be. For instance, if you are studying the evolution of cooperation, you might select 'Cooperate' and 'Defect' as your strategies and the Prisoner's dilemma as your pay-off matrix, but maybe you will pick something more general. Unfortunately, in most of EGT a clear distinction between genotype and phenotype is not drawn, and they are usually equated. At the end of this step you have a game matrix G. Sometimes when mutation or innovation are explicitly necessary even in the inviscid model, further analytic approaches are taken at this stage. I recommend Hofbauer &amp; Sigmund (2003) for a broad treatment of step 1.
Now you need a basic intuition of what the 'default' behavior in this interaction is, so solve the replicator dynamics of G.
The main interest in EGT right now is structured populations. This is where computational modeling is typically used. However, before I turn to simulation, I first try the best analytic approach I know. I use the Ohtsuki-Nowak transform on G to analytically solve the interaction for random graphs (Ohtsuki &amp; Nowak, 2006).
If I am still interested in the question, and steps 2 and 3 didn't capture all the subtlety of the system I want to study, then I start building a multi-agent computational model. I make sure that my model has some way to scale to the completely inviscid case of replicatory dynamics, and the simply-structured case of ON-transform. If my computational model disagrees with the analytic approach in these regimes, then I become worried. Otherwise, I continue with standard agent-based modeling techniques. Personally, I code in Matlab. I have never seen a modern EGT simulation that required the performance of C/Fortran. As suggested in another answer, if you don't have a programming background then you can use NetLogo. However, my experience is that the models are typically simple enough to implement from scratch, and NetLogo models typically hide from you some very important subtleties (like which reproduction rule to use: death-birth, birth-death, immitation?) and usually lead to weaker papers. 


Note the broad theme. These models are typically described as differential equations, and this approach is preferred to agent-based models. However, if a clean differential equations approach does not capture all the subtleties of what you are studying, then an ABM paradigm is adopted.